Power, Roots And Radicals: Definition & Examples

Q: What is an example of solving powers and radicals?

8 1/2 = √8 = √4·2 = √4·√2 = 2√2

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If you need to revert the process to find out what number was raised to a given power, then you can use roots, also known as radicals. In this article, we will define what powers and roots are, and we will explain in more detail how they are related to each other, their rules and properties, and some practical examples of how to solve them.

Powers Definition

Power is the exponent that a variable is raised to. For example, the expression x² is read as "x to the power of 2", or "x squared", which means that the value of x is multiplied by itself as many times as the value of the power or exponent. In this case, x is multiplied by itself two times. LOLA

Examples of Powers

If the value of x is 5, then we can calculate $x^{2}$ like this,

$x^{2} = 5^{2} = 5 \cdot 5 = 25$ .

Likewise, we can calculate $x^{3}$ and $x^{4}$ :

$x^{3} = 5^{3} = 5 \cdot 5 \cdot 5 = 125$ .

Notice that if you already know the value of 5², which is 25, you can multiply it by 5 one more time to obtain the value of 5³.

$x^{4} = 5^{4} = 5 \cdot 5 \cdot 5 \cdot 5 = 5^{3} \cdot 5 = 125 \cdot 5 = 625$ .

Important to remember

If a variable has no power or exponent, then it is assumed to be 1. For example, $x^{1} = x$ .

Also, any variable to the power of 0 (zero) equals 1. By convention, we have $x^{0} = 1$ .

Rules and Properties of Powers

You can refer to Powers and Exponents for a more detailed explanation of the rules you need to use when working with powers.

Here are the rules and properties of powers, also known as exponent properties or laws, that you need to keep in mind.

Tip: Easy-to-remember property name	Property expression
Same-base product	$x^{a} \cdot x^{b} = x^{a + b}$
Same-base division	$x^{a} \div x^{b} = x^{a - b}$
Same-exponent product	$x^{a} \cdot y^{a} = {(x y)}^{a}$
Same-exponent division	$x^{a} \div y^{a} = {(x \div y)}^{a}$
Double exponent	${(x^{a})}^{b} = x^{a \times b}$
Zero exponent	$x^{0} = 1$
Negative exponent	$x^{- a} = \frac{1}{x^{a}}$
Fractional exponent	$x^{\frac{a}{b}} = \sqrt[b]{x^{a}}$

Smart exponent properties - StudySmarter

Roots and Radicals Definition

Roots, also known as radicals, are the inverse of powers. To calculate the $n^{t h}$ root of a number $(\sqrt[n]{x})$ , we need to find what number multiplied by itself n times gives us the number inside the radical symbol (x) that is called the radicand.

Powers Roots And Radicals Indicating the nth root of a number StudySmarter The nth root of a number x

Examples of Roots and Radicals

The index n of a root can be any positive integer, and it gives the name to the radical. Let's explore some examples of the most common roots and radicals that you will find in Math.

Square Roots

If $n = 2$ , $\sqrt[2]{x}$ refers to the square root of a number x. In this case, we normally omit the 2, and simply write $\sqrt{x}$ .

If you want to find the square root of a number, you need to find out what number times itself would give us the number inside the square root.

If you want to find the square root of 25, you need to find what number multiplied by itself equals 25.

$\sqrt{25} = \pm 5$

But why is the result ± 5?

This is because both 5 and -5 when raised to the power of 2, give 25:

5 \cdot 5 = 25, (- 5) \cdot (- 5) = {(- 1)}^{2} \cdot (5^{2}) = 25 .

Therefore, there are always two answers when we take the square root of a positive number.

$\sqrt{- 25} \neq - 5$

The square root of a negative number has no real solution; imaginary numbers are required in this case. Only positive numbers can have their square root taken in this way because we cannot find any number once raised to power 2 that would give us a negative number.

Types of Square Roots

Square roots can be classified according to the type of number inside the root, as follows.

The square root of perfect squares:

The square root of perfect squares gives an integer as a result. It is very easy to calculate and useful to remember when working with expressions containing powers and roots. It helps to evaluate and simplify these types of expressions. Just as a reminder, here are the first ten.

$\sqrt{1}$	$\sqrt{4}$	$\sqrt{9}$	$\sqrt{16}$	$\sqrt{25}$	$\sqrt{36}$	$\sqrt{49}$ $\sqrt{49}$	$\sqrt{64}$	$\sqrt{81}$	$\sqrt{100}$
$\pm 1$	$\pm 2$	$\pm 3$	$\pm 4$	$\pm 5$	$\pm 6$	$\pm 7$	$\pm 8$	$\pm 9$	$\pm 10$

The square root of numbers that are not perfect squares:

The square root of numbers that are not perfect squares is not an integer. They produce irrational numbers with infinite decimals. To represent this type of number more exactly, they are left in their root form.

$\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7}$ .

If the number inside the root has a square number as a factor, then it can be simplified.

$\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2}$

The steps to follow to simplify radicals are:

Write the number inside the root as the multiplication of its factors. One of the factors should be a square number.

$\sqrt{8} = \sqrt{4 \cdot 2}$

Split the factors into separate roots

$\sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2}$

Simplify the terms

$\sqrt{4} \cdot \sqrt{2} = 2 . \sqrt{2}$

Take out the multiplication symbol

$2 \cdot \sqrt{2} = 2 \sqrt{2}$

Cube Roots

If $n = 3$ , $\sqrt[3]{x}$ refers to the cube root of a number x.

If you want to find the cube root of a number, you need to find out what number multiplied by itself three times would give us the number inside the cube root. It is the opposite of raising a number to the $3^{r d}$ power.

If you want to find the cube root of 8, you need to find what number multiplied by itself three times equals 8.

\sqrt[3]{8} = 2

, in fact

2 \cdot 2 \cdot 2 = 8 .

Notice that in this case, we have only one answer, not two. This is because when you multiply a negative number by itself three times, the result is also negative.

$(- 2) \cdot (- 2) \cdot (- 2) = - 8$ .

Therefore, the only possible answer is 2. $\sqrt[3]{- 8} = - 2$

$\sqrt[3]{- 8} = - 2$

Cube roots CAN take the cube root of a negative number.

Examples of Roots and Radicals: Other Roots

4^th Root: The rules are similar to the ones from square roots.
5^th Root: The rules are similar to cube roots.
In general terms, odd roots have one solution, and even roots have two solutions.

Rules and Properties of Roots and Radicals

When working with roots and radicals, you need to remember the following rules and properties:

Multiplying Radicals

As long as the index of the roots is the same, you can multiply radicals with different numbers inside the root by simply combining them into one root and multiplying the numbers inside the root. Likewise, you can split a root into separate roots using factors.

$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ .

$\sqrt{2} \cdot \sqrt{5} = \sqrt{2 \cdot 5} = \sqrt{10}$ .

Dividing Radicals

Similarly, as long as the index of the roots is the same, you can divide radicals with different numbers inside the root by combining them into one root and dividing the numbers inside the root.

$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} = \sqrt{a \div b}$ .

$\frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}} = \sqrt{10 \div 2} = \sqrt{5}$ .

Multiplying a Square Root by Itself

If you multiply the square root of a number by itself, you should obtain the original value.

$\sqrt{a} \cdot \sqrt{a} = {(\sqrt{a})}^{2} = a$ .

a·a=a2=a

$\sqrt{3} \cdot \sqrt{3} = {(\sqrt{3})}^{2} = 3$ .

Multiplying a Number by a Radical

When multiplying a number by a radical, the order of the factors does not matter, and the result should be the number followed by the radical.

$a \cdot \sqrt{b} = \sqrt{b} \cdot a = a \sqrt{b}$ .

$3 \cdot \sqrt{2} = \sqrt{2} \cdot 3 = 3 \sqrt{2}$ .

Adding or Subtracting Radicals

To add or subtract radicals, the number inside the roots must be the same. You add or subtract the numbers outside the root.

$a \sqrt{d} + b \sqrt{d} = (a + b) \sqrt{d}$ .

$a \sqrt{d} - b \sqrt{d} = (a - b) \sqrt{d}$ .

$5 \sqrt{3} + 2 \sqrt{3} = (5 + 2) \sqrt{3} = 7 \sqrt{3} 5 \sqrt{3} - 2 \sqrt{3} = (5 - 2) \sqrt{3} = 3 \sqrt{3}$

To add or subtract radicals, you might need to simplify them first to find like terms.

You cannot add $\sqrt{2} + \sqrt{8}$ , but you can simplify $\sqrt{8}$ first,

\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2}

Then you can solve $\sqrt{2} + \sqrt{8} = \sqrt{2} + 2 \sqrt{2} = 3 \sqrt{2}$ .

Multiplying Brackets Containing Radicals

To multiply brackets containing radicals, each term in the first bracket must be multiplied by each term in the second bracket. Then you can combine like terms.

(2 + \sqrt{3}) (5 + \sqrt{3}) = 2 \cdot 5 + 2 \sqrt{3} + 5 \sqrt{3} + {(\sqrt{3})}^{2}

$= 10 + 7 \sqrt{3} + 3$

$= 13 + 7 \sqrt{3}$

How Do You Write Powers as Roots and Roots as Powers?

To write powers as roots and roots as powers, we need to understand how fractional exponents work.

Fractional Exponents

Fractional exponents are equivalent to roots as shown in the following law of exponents.

$x^{\frac{a}{b}} = \sqrt[b]{x^{a}}$ .

Using this expression, you can write any fractional exponent as a root.

$x^{\frac{1}{2}} = \sqrt{x}$

$x^{\frac{1}{3}} = \sqrt[3]{x}$

$x^{\frac{2}{3}} = \sqrt[3]{x^{2}}$

You can use the same expression to write any root as a fractional exponent.

$\sqrt[4]{x} = x^{\frac{1}{4}}$

$\sqrt[6]{x^{5}} = x^{\frac{5}{6}}$

Read the Rational Exponents explanation to learn more about this topic.

Solving Powers, Roots and Radicals

Now that you know how to work with fractional exponents and, keeping in mind the laws of exponents, you have everything you need to evaluate or simplify expressions containing powers, roots and radicals. Here are some examples.

a) Evaluate or simplify $\sqrt{50}$

Remembering perfect squares, you can change $\sqrt{50}$ to $\sqrt{25 \cdot 2}$

$\sqrt{50} = \sqrt{25} \cdot \sqrt{2}$

$\sqrt{50} = 5 \sqrt{2}$

$5 \sqrt{2}$ cannot be simplified further, so it is left in its square root form.

b) Evaluate or simplify $\frac{\sqrt{x} \cdot \sqrt[4]{x}}{\sqrt[3]{x}}$

$\frac{\sqrt{x} \cdot \sqrt[4]{x}}{\sqrt[3]{x}} = \frac{x^{\frac{1}{2}} . x^{\frac{1}{4}}}{x^{\frac{1}{3}}}$ Transforming the roots into fractional exponents.

$\frac{\sqrt{x} . \sqrt[4]{x}}{\sqrt[3]{x}} = \frac{x^{\frac{3}{4}}}{x^{\frac{1}{3}}}$ Using the law of exponents $x^{a} \cdot x^{b} = x^{a + b}$ .

$\frac{\sqrt{x} \cdot \sqrt[4]{x}}{\sqrt[3]{x}} = x^{\frac{3}{4} - \frac{1}{3}}$ Using the law of exponents $x^{a} \div x^{b} = x^{a - b}$ .

$\frac{\sqrt{x} \cdot \sqrt[4]{x}}{\sqrt[3]{x}} = x^{\frac{5}{12}}$

c) Evaluate or simplify $\frac{12 x^{4} y^{2}}{3 x^{6}}$

$\frac{12 x^{4} y^{2}}{3 x^{6}} = 4 x^{- 2} y^{2}$ Using the law of exponents $x^{a} \div x^{b} = x^{a - b}$ , and simplifying $\frac{12}{3}$ .

$\frac{12 x^{4} y^{2}}{3 x^{6}} = \frac{4 y^{2}}{x^{2}}$ Using the law of exponents $x^{- a} = \frac{1}{x^{a}}$ .

d) Evaluate or simplify ${(\frac{x y^{2}}{x^{3}})}^{- 3}$

${(\frac{x y^{2}}{x^{3}})}^{- 3} = {(\frac{x^{3}}{x y^{2}})}^{3}$ Using the law of exponents $x^{- a} = \frac{1}{x^{a}}$ flips the fraction,

$= \frac{{(x^{3})}^{3}}{{(x y^{2})}^{3}}$ Distributing the exponent into the numerator and denominator,

$= \frac{x^{9}}{x^{3} y^{6}}$ Using the law of exponents $x^{a} \div x^{b} = x^{a - b}$ ,

=x9x3y6 ${(\frac{x y^{2}}{x^{3}})}^{- 3} = \frac{x^{6}}{y^{6}}$

Powers, Roots And Radicals - Key takeaways

Power is the exponent that a variable or number is being raised to.
Roots or radicals are the inverses of powers.
Odd roots will have one solution, while even roots will have two.
Only positive numbers can have their square roots taken without using imaginary numbers.
Negative numbers can have their cube roots taken.
Knowing the square roots of perfect squares and the laws of exponents is very useful when evaluating or simplifying algebraic expressions containing powers and roots.

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Frequently Asked Questions about Powers Roots And Radicals

What is the power of a radical?

The law of fractional exponents states that x to the power of a/b is equal to the b^th root of x^a.

What is the relationship between powers and radicals?

A fractional exponent can be written as a root or radical, and vice versa. The law of fractional exponents is used to write powers as a root. This law states that x to the power of a/b is equal to the b^th root of x^a.

How do you determine powers and radicals?

To calculate powers, the number or variable is multiplied by itself as many times as the value of the power or exponent.

To calculate radicals, you will be looking for numbers when raised to a given power would give you the initial number – the one where you are searching for its root.

What is an example of solving powers and radicals?

8^1/2 = √8 = √4·2 = √4·√2 = 2√2

How do you evaluate powers and radicals?

If you have a number raised to a fractional exponent, i.e. 8^2/3, you can evaluate this in two ways:

Calculate the power first, and then take the cube root of the result: 8^2/3 =(8²)^1/3= ∛64 = 4.
Calculate the cube root first, and then do the power: 8^2/3 = (8^1/3)² = (∛8)² = 2² = 4.

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Powers Roots And Radicals

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Powers Definition

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Roots and Radicals Definition