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Powers and Roots

Powers and Roots

Powers are used when we want to multiply a number by itself repeatedly.

What are powers?

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.

If the value of x is 5, then we can calculate x² like this:

Likewise, we can calculate and :

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.

Important to remember:

If a variable has no power or exponent, then it is assumed to be 1. For example,

Also, any variable to the power of 0 (zero) equals 1. For example,

You can refer to Exponential Rules for a more detailed explanation of the rules you need to use when working with exponents.

Just as a reminder, these are the exponential rules that you need to keep in mind:

What are roots?

Roots are the inverse of powers. To calculate the root of a number , we need to find what number multiplied by itself n times, gives us the number inside the radical symbol (x).

1. Square root

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.

But why is the result ± 5?

This is because the square root of 25 can be either 5 or -5.

5 x 5 = 25

(-5) x (-5) = 25

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

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.

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:

± 1± 2± 3± 4± 5± 6± 7± 8± 9± 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 and called surds. For example: .

If the number inside the root of a surd has a square number as a factor, then it can be simplified. For example: .

You can read about Surds for more detail.

2. Cube Root

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

If you want to find the cube root of 8, you need to find what number multiplied by itself 3 times equals 8.

Notice that in this case, we have only one answer, not two. This is because when you multiply a negative number by itself 3 times, the result is also negative.

(-2) x (-2) x (-2) = -8

Therefore, the only possible answer is:

2 x 2 x 2 = 8

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

3. Other roots

  • 4th Root: The rules are similar to the ones from square roots.

  • 5th Root: The rules are similar to cube roots.

  • In general terms, odd roots have one solution, and even roots have two solutions.

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 exponential rule:

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

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

Evaluating and simplifying expressions with powers and roots

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

Example 1

Evaluate or simplify

Remembering perfect squares, you can change to

is an example of a surd because it cannot be simplified further, so it is left in its square root form. Remember to read more about Surds!

Example 2

Evaluate or simplify

transforming the roots into fractional exponents

using the exponential rule

using the exponential rule

Example 3

Evaluate or simplify

using the exponential rule

using the exponential rule

Example 4

Evaluate or simplify

using the exponential rule flip the fraction

distributing the exponent into the numerator and denominator

using the exponential rule

Powers and Roots - Key takeaways

  • Power is the exponent that a variable or number is being raised to.

  • The root is the opposite of power.

  • 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 exponential rules is very useful when evaluating or simplifying algebraic expressions containing powers and roots.

Frequently Asked Questions about Powers and Roots

Power is the exponent that a variable or number is being raised to, which in practice means that the number or variable is multiplied by itself as many times as the value of the power or exponent. Roots are the opposite, they find what number multiplied by itself n times equals the number inside the root, where n is the index of the root.

The exponential rule of fractional exponents is used to write powers as a root, which means x to the power of a over b is equal to the bth root of x to the power of a.

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

To simplify roots and powers it is useful to know the square root of perfect squares and the exponential rules.

Final Powers and Roots Quiz

Question

What are powers?

Show answer

Answer

Power is the exponent that a variable or number is being raised to, which in practice means that the number or variable is multiplied by itself as many times as the value of the power or exponent.

Show question

Question

What are roots?


Show answer

Answer

Roots are the opposite of powers, they find what number multiplied n times equals the number inside the root, where n is the index of the root.

Show question

Question

How many solutions do odd roots have?

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Answer

One solution

Show question

Question

How many solutions do even roots have?

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Answer

Two solutions

Show question

Question

Calculate \(2^5\).

Show answer

Answer

\(32\).

Show question

Question

Evaluate \(\sqrt{243}\).

Show answer

Answer

\(9\sqrt{3}\).

Show question

Question

Write \(x^{\frac{5}{4}}\) as a root.

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Answer

\(\sqrt[4]{x^5}\).

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Question

Write \(\sqrt[3]{x^8}\) as a power.

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Answer

\(x^{\frac{8}{3}}\).

Show question

Question

Evaluate or simplify \((3x^3y)^2\).

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Answer

\(9x^6y^2\).

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Question

Evaluate or simplify \[\frac{x^{2y+1}}{x^{2y-1}}.\]

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Answer

\(x^2\).

Show question

Question

What are surds?

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Answer

Surds are expressions that contain a square root, cube root or other roots, which produce an irrational number as a result, with infinite decimals. They are left in their root form to represent them exactly.

Show question

Question

How do you multiply brackets containing surds?


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Answer

To multiply brackets containing surds, each term in the first bracket must be multiplied by each term in the second bracket.

Show question

Question

How do you simplify surds?


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Answer

The steps to simplify surds are:

  • Write the number inside the root as the multiplication of its factors. One of the factors should be a square number 
  • Split the factors into separate roots 
  • Simplify the terms
  • Take out the multiplication symbol

Show question

Question

How do you rationalise the denominator of surds?

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Answer

  • If the denominator is a surd, then multiply the numerator and denominator by that surd.
  • If the denominator has two terms, one rational and a surd, then multiply the numerator and denominator by the expression conjugate of the denominator.

Show question

Question

What is the correct simplification of \(\sqrt{20}\)?

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Answer

\(2\sqrt{5}\).

Show question

Question

Simplify \(\sqrt{300}\).

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Answer

\(10\sqrt{3}\).

Show question

Question

Simplify \(\sqrt{243}\).

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Answer

\(9\sqrt{3}\).

Show question

Question

Which of these is equivalent to \(7\sqrt{2}\)?

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Answer

\(sqrt{98}\).

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Question

Simplify \(\sqrt{32}\).

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Answer

\(4\sqrt{2}\).

Show question

Question

How would you rationalize the fraction \[ \frac{4}{\sqrt{2}}?\]

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Answer

By multiplying the fraction by \(\frac{\sqrt{2}}{\sqrt{2}} \).

Show question

Question

How can you simplify \[ \frac{ \sqrt{5} + 3}{\sqrt{5} - 2}?\]

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Answer

By multiplying both the numerator and denominator by \(\sqrt{5} + 2 \).

Show question

Question

Can you add \(\sqrt{2}\) and \(\sqrt{3}\)?

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Answer

No, the numbers inside the square roots are not the same.

Show question

Question

Can you add \(\sqrt{2}\) and \(\sqrt{50}\)?

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Answer

Yes, if you simplify the \(\sqrt{50}\) first.

Show question

Question

What is the multiplication law?

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Answer

Add the powers.

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Question

What is the division law?

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Answer

Divide the powers.

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Question

How do you calculate a power to a power?


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Answer

Multiply the powers.

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Question

How do you calculate a power of 0?


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Answer

Any number to the power of 0 is equal to 1.

Show question

Question

What does a negative power do?


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Answer

It takes the reciprocal of a number.

Show question

Question

Which of these is the correct way to write \(\sqrt[7]{x^2}\) with powers?

Show answer

Answer

\(x^{\frac{7}{2}}\).

Show question

Question

Which of these is the correct way to write \(x^{\frac{5}{9}}\) with roots?

Show answer

Answer

\(\sqrt[9]{x^5}\).

Show question

Question

To calculate \(6^4\) how many times should you multiple \(6\) by iteslf?

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Answer

\(4\).  

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Question

Which of these is the correct equivalent for \(x^0\)?

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Answer

1.

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Question

Which of these numbers is a perfect square?

Show answer

Answer

\(36\).

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