Property	Rule	Description	Example
Product of Powers	am⋅an = am+n	Multiplication of same bases, add exponents.	23⋅24 = 23+4 = 27= 128
Quotient of Powers	am/an​=am−n	Dividsion of same bases, subtract exponents.	56/52 = 56−2 =54 = 625
Power of a Power	(am)n=am⋅n	Raise an expression to another power, multiply the exponents.	(32)3 = 32⋅3 = 36 = 729
Power of a Product	(ab)n=an⋅bn	Raise a product to a power, raise each factor to the power separately.	(2⋅4)3 = 23⋅43 = 8⋅64 = 512
Power of a Quotient	(a/b​)n=an​ /bn	Raise a quotient to a power, raise both numerator and denominator to the power.	(3/2)2 = 32/22 = 9/4​
Zero Exponent	a0 = 1	Any non-zero base raised to the power of zero is 1.	70 = 1
Negative Exponent	a−n = 1/an	Negative exponent indicates reciprocal of the base.	4-2 = 1/42 = 1/16
Fractional Exponent	am/n= [Tex]\sqrt[n]{a^m}[/Tex]​	Fractional exponent represents a root. Numerator is power, denominator is root.	81/3 = [Tex]\sqrt[3]{8}[/Tex] = 2

Practice Questions on Exponents and Powers – Solved

These Practice Questions on Exponents and Powers will help you better understand and apply the concept.

Example 1. Simplify the expression 3⁴⋅3²

According to product of powers property: a^m⋅ aⁿ= a^{m + n}

Therefore, 3⁴⋅3²= 3⁴⁺²

= 3⁶

= 3×3×3×3×3×3

= 729

Example 2: Simplify the expression 10⁷/10³

According to quotient of powers property: a^m/aⁿ​= a^m−n

Therefore, 10⁷/10³ = 10^7-3

= 10⁴

= 10,000

Exampe 3: Simplify the expression (2⁵)²

According to power of a power property: (a^m)ⁿ= a ^m⋅n

Therefore, (2⁵)² = 2^5·2

= 2¹⁰

= 1024

Example 4: Simplify the expression 7⁻³

According to negative exponent property: a⁻ⁿ= 1/aⁿ

Therefore, 7 ^-3 = 1/7³

= 1/343​

Example 5: Simplify the expression: (4² × 4^-3).

Using the product rule a^m × aⁿ= a^m+n

Now, by applying the rule in given expression 4²×4^-3 we get,

= 4^2+(-3)

= 4^-1

Now the outcome is in the form of negative exponent rule a^-n = 1/aⁿ.

∴ 4^-1 =1/4¹

Therefore, the simplified expression is 1/4

Example 6: Simplify the expression: (a³b²)³

This questions involves the power of a product rule (ab)ⁿ and the power of a power rule (a^m)ⁿ.

First we will apply the power of a product rule:

Distribute the outer exponent to both a³ and b².

In the given expression (a³b²)³

(a³b²)³=(a³)³×(b²)³

Now, apply the power of a power rule to each term:

⇒ (a³)³=a^3×3=a⁹

⇒ (b²)³=b^2×3=b⁶

Combining the results we get,

(a³b²)³=a⁹× b⁶

Example 7: Simplify the expression: [Tex]\left(\frac{2^3}{5}\right)^2[/Tex]

2³=2×2×2

=8

Rewrite the expression with the simplified numerator

(8/5)²

= 8²/5²

8² = 64

5² = 25

We can write it as = 64/25

Example 8: Solve for y: 5^2y+1 = 125

Given, 5^2y+1 = 125

We know, 125 = 5³

Hence, 5^2y+1 = 5³

Therefore, 2y+1 = 3

= 2y = 3-1 = 2

= 2y = 2

Hence y = 1

Example 9: [Tex]Solve: ( \,\frac{a^4b^2}{a^{-1}b^3}) \,^{-2}[/Tex]

[Tex]Given, ( \,\frac{a^4b^2}{a^{-1}b^3}) \,^{-2}[/Tex]

On solving powers, we get

(a^4-(-1) b ^-2-3)^-2

= (a⁵b^-5)^-2

= a^-10 b¹⁰

= b¹⁰/a¹⁰

= (b/a)¹⁰

Example 10: Evaluate (8^1/3 × 16^1/4)

We have, (8^1/3 × 16^1/4)

By Fractional Exponent Property we have,

8^1/3 = [Tex]\sqrt[3]{8} = 2[/Tex]

and 16^1/4 = [Tex]\sqrt[4]{16} = 2[/Tex]

Hence the given equation becomes:

8^1/3 × 16^1/4 = 2 × 2 = 4

Read More:

• Adding and Subtracting Exponents
• Laws of Exponents
• How to Multiply and Divide Exponents
• Fractional Exponents

Practice Questions on Exponents and Powers – Unsolved

1. Simplify the expression: 5³×5².

2. Simplify the expression: [Tex]\frac{8^5}{8^2}[/Tex]

3. Simplify the expression: (3⁴)².

4. Simplify the expression: (2×7)³.

5. Simplify the expression: (4/5)².

6. Simplify the expression: 9⁰.

7. Simplify the expression: 7⁻³.

8. Simplify the expression: (x²×y³)⁴.

9. Simplify the expression: [Tex]\frac{6^3 \times 6^2}{6^4}[/Tex]

10. Simplify the expression: 10³×2³.

Practice Questions on Exponents and Powers – FAQs

What are the 5 properties of exponential functions?

The 5 properties of exponential functions are:

• Exponential functions have graphs that are continuous and smooth.
• The horizontal asymptote is y = 0.
• The domain is all real numbers, and the range is positive real numbers (y > 0).
• If the base b > 1, the function shows exponential growth. If 0 < b < 1, the function shows exponential decay.
• The y-intercept is at (0,a) where (x)=a⋅b^x
  

What are the 7 rules of exponents?

The 7 rules of exponents are:

• Product of Powers
• Quotient of Powers
• Power of a Power
• Power of a Product
• Power of a Quotient​
• Zero Exponent
• Negative Exponent

What are the 5 exponential laws?

The 5 exponential laws are:

• a^m⋅aⁿ= a ^{m + n}
• a^m/aⁿ​ = a^m−n
• (a^m)ⁿ= a ^m⋅n
• a⁰ = 1
• a^-n= – (1/aⁿ)​

What is the power of 0 rule?

The power of 0 rule states that any non-zero number raised to the power of zero is equal to 1.

?⁰=1 (for any a≠0).

How to solve exponent questions?

To solve exponent questions, the steps are as follows:

Step 1: Determine which law of exponents applies to the given problem.

Step 2: Apply the relevant exponent rule to simplify the expression.

Step 3: If the problem involves multiple terms, combine them using the appropriate laws.

Step 4: Calculate the final value if necessary.

What is the meaning of the base and exponent in an expression?

In an expression of the form aⁿ:

• Base (a): The number that is being multiplied.
• Exponent (n): The number that indicates how many times the base is multiplied by itself.