Powers and Roots

Powers and roots are basic math concepts that help us write numbers in a simpler way. Powers show how many times a number is multiplied by itself, while roots do the opposite—they find the number that was multiplied to get a given value. These ideas make calculations easier and are used often in algebra and higher mathematics.

Powers (Exponents)

Powers (exponents) are a way to show how many times to multiply a number by itself. We write it like this: aⁿ. Here’s what it means:

• a is the number we start with (called the base).
• n is how many times we multiply it (called the exponent).

For example:

3³= 3 × 3 ×3 = 27

The general form is written as aⁿ, where:

• a is the base
• n is the exponent

Properties of Exponents

• Product of Powers: This simply means that when multiplying two powers of the same base, then we add the exponent to the result.

a^m \times a^n = a^{m+n}

• Power of a Power: For applications of a power to another power, multiply the power by the base.

(a^m)^n = a^{m \times n}

• Quotient of Powers: When dividing two powers with the same base, subtract the exponents.

\frac{a^m}{a^n} = a^{m-n}

• Negative Exponents: A negative exponent is an indication of taking the reciprocal of the base raised to the positive exponent of equal value

a^{-m} = \frac{1}{a^m}

Positive and Negative Powers

Positive powers are straightforward repeated multiplication.

For Example: 2⁴ = 2 × 2 × 2 × 2 = 16.

Negative powers indicate the reciprocal of the base raised to the positive exponent.

Example: 2⁻³ = 1/ 2³ = 1/8

Read more about fractional exponents.

Roots

Roots represent the inverse operation of raising a number to a power.

Specifically, taking the root of a number is the process of finding a value that, when raised to a certain exponent, equals the given number. If x² = 16, then the square root of 16 is the number that, when squared, results in 16. That is,

√16 = 4

Square, Cube, and Nth Roots

• Square Root (√): The square root of a number is the value that, when squared, gives the original number.

Example: √25 = 5 because 5² = 25.

• Cube Root (∛): The cube root of a number is the value that, when cubed, gives the original number.

Example: ∛27 = 3 because 3³ =27.

• Nth Root: This is a general form where the root is based on the value of n.

Example: ⁴√16 = 2 because 2⁴ = 16.

Relationship Between Powers and Roots

Roots are essentially fractional exponents. The square root of 'a' can be written as '(a,' the cube root as 'a,' and so on. For instance:

a^{1/2} = \sqrt{a}, \quad a^{1/3} = \sqrt[3]{a}

How Do You Convert Roots to Powers?

To convert roots to powers, express the root as a fractional exponent:

• \sqrt{a} = a^{1/2}
• \sqrt[3]{a} = a^{1/3}
• \sqrt[n]{a} = a^{1/n}

Solved Problems on Powers and Roots

Question 1: Simplify (3) × 3⁻⁴.

Solution:

Using the power of a power rule:

(3²)³ = 3^2×3 = 3⁶.

Now using the product of powers rule:

3⁶ × 3⁻⁴ = 3⁶⁻⁴ = 3² =9.

Question 2: Simplify √36 × √ 4.

Solution:

Using the product of roots rule:

√36 × √4 = √36×4 = √144 = 12.

Question 3: Evaluate 16.

Solution:

First, express 16 as a power of 2: 16=2⁴

Now apply the fractional exponent: 16^3/4

=(2⁴) ^3/4

= 2^4×3/4 = 2³ = 8.

Question 4: Simplify 2⁻³.

Solution:

Using the negative exponent rule:

2 ⁻³ = 1 / 2³ = 1 / 8.

Question 5: Simplify \sqrt[3]{64}.

Solution:

Express 64 as a power of 4:

64 = 4³.

Now, take the cube root:

\sqrt[3]{64} = \sqrt[3]{4^3} = 4

Practice Problems on Powers and Roots

Question 1: Simplify (2⁴ × 2³) ²÷ 2⁶

Question 2: Evaluate: \frac{7^4}{7^2}.

Question 3: Find the Value of: (√81)²

Question 4: Simplify the Root Expression: \sqrt[4]{16 \times 64}.

Question 5: Evaluate the Following: \frac{\sqrt[3]{3}}{\sqrt[3]{27}}.

Question 6: Find the Value of: \left(2^{\frac{2}{3}}\right)^4.

Question 7: Simplify the Expression: √ 50 ✕ √ 2.

Question 8: Evaluate the Following: \frac{2^5}{2^2}

Question 9: Find the Value of: \sqrt[5]{32^2}

Question 10: Simplify the Root Expression: \sqrt{\frac{36}{9}}.

Answer Key

1. 256
2. 49
3. 81
4. 8
5. 1/3^-2/3
6. 6.3496
7. 10
8. 8
9. 4
10. 2