Exponent Rules Practice Problems - Master Powers & Bases

Practice exponent rules with step-by-step problems. Master multiplication, division, power of powers, negative exponents, and zero exponents through guided exercises.

📚Master Exponent Rules Through Interactive Practice
  • Apply multiplication rule: a^n × a^m = a^(n+m) with same bases
  • Solve division problems using a^n ÷ a^m = a^(n-m) formula
  • Master power of powers rule: (a^n)^m = a^(n×m) in complex expressions
  • Work with negative exponents using a^(-n) = 1/a^n conversions
  • Practice zero and one exponent rules: a^0 = 1 and 1^n = 1
  • Solve multi-step problems combining all exponent rules

Understanding Exponents Rules

Complete explanation with examples

Definition of Exponentiation

Exponents are a way to write the multiplication of a term by itself several times in a shortened form.

The number that is multiplied by itself is called the base, while the number of times the base is multiplied is called the exponent.

A - Exponentiation

an=aaa a^n=a\cdot a\cdot a ... (n times)

For example:

5555=54 5\cdot5\cdot5\cdot5=5^4

5 5 is the base, while 4 4 is the exponent.

In this case, the number 5 5 is multiplied by itself 4 4 times and, therefore, it is expressed as 5 5 to the fourth power or 5 5 to the power of 4 4 .

Detailed explanation

Practice Exponents Rules

Test your knowledge with 78 quizzes

Simplify the following equation:

\( 7^3\times5^2\times7^4\times5^3= \)

Examples with solutions for Exponents Rules

Step-by-step solutions included
Exercise #1

(23)6= (2^3)^6 =

Step-by-Step Solution

To solve the given expression (23)6 (2^3)^6 , we apply the power of a power rule (am)n=amn (a^m)^n = a^{m \cdot n} . Here, a=2 a = 2 , m=3 m = 3 , and n=6 n = 6 .

Thus, we calculate the exponent:

36=18 3 \cdot 6 = 18

So, (23)6=218 (2^3)^6 = 2^{18} .

Answer:

218 2^{18}

Exercise #2

1120=? 112^0=\text{?}

Step-by-Step Solution

We use the zero exponent rule.

X0=1 X^0=1 We obtain

1120=1 112^0=1 Therefore, the correct answer is option C.

Answer:

1

Video Solution
Exercise #3

50= 5^0=

Step-by-Step Solution

We use the power property:

X0=1 X^0=1 We apply it to the problem:

50=1 5^0=1 Therefore, the correct answer is C.

Answer:

1 1

Video Solution
Exercise #4

Solve the following problem:

(34)×(32)= \left(3^4\right)\times\left(3^2\right)=

Step-by-Step Solution

In order to solve this problem, we'll follow these steps:

  • Step 1: Identify the base and exponents

  • Step 2: Use the formula for multiplying powers with the same base

  • Step 3: Simplify the expression by applying the relevant exponent rule

Now, let's work through each step:

Step 1: The given expression is (34)×(32) (3^4) \times (3^2) . Here, the base is 3, and the exponents are 4 and 2.

Step 2: Apply the exponent rule, which states that when multiplying powers with the same base, we add the exponents:
am×an=am+n a^m \times a^n = a^{m+n}

Step 3: Using the rule identified in Step 2, we add the exponents 4 and 2:
34×32=34+2=36 3^4 \times 3^2 = 3^{4+2} = 3^6

Therefore, the simplified form of the expression is 36 3^6 .

Answer:

36 3^6

Video Solution
Exercise #5

Reduce the following equation:

(32)4×(53)5= \left(3^2\right)^4\times\left(5^3\right)^5=

Step-by-Step Solution

To solve this problem, we'll employ the power of a power rule in exponents, which states that (am)n=am×n(a^m)^n = a^{m \times n}.

Let's apply this rule to each part of the expression:

  • Step 1: Simplify (32)4(3^2)^4
    According to the power of a power rule, this becomes 32×4=383^{2 \times 4} = 3^8.

  • Step 2: Simplify (53)5(5^3)^5
    Similarly, apply the rule here to get 53×5=5155^{3 \times 5} = 5^{15}.

After simplifying both parts, we multiply the results:

38×5153^8 \times 5^{15}

Thus, the reduced expression is 38×515\boxed{3^8 \times 5^{15}}.

Answer:

38×515 3^8\times5^{15}

Video Solution

Frequently Asked Questions

How do you multiply exponents with the same base?

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When multiplying powers with the same base, add the exponents: a^n × a^m = a^(n+m). For example, 5^2 × 5^3 = 5^(2+3) = 5^5. The base stays the same, only the exponents are added together.

What is the rule for dividing exponents with the same base?

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When dividing powers with the same base, subtract the exponents: a^n ÷ a^m = a^(n-m). For example, 7^5 ÷ 7^2 = 7^(5-2) = 7^3. Remember the base cannot be zero when using this rule.

How do you solve power of a power problems?

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For power of a power, multiply the exponents: (a^n)^m = a^(n×m). For example, (3^2)^4 = 3^(2×4) = 3^8. This rule applies when you have parentheses around a base with an exponent, and another exponent outside.

What happens when you raise a number to the power of zero?

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Any non-zero number raised to the power of zero equals 1: a^0 = 1. For example, 5^0 = 1, (-3)^0 = 1, and 100^0 = 1. However, 0^0 is undefined in mathematics.

How do you work with negative exponents?

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A negative exponent means taking the reciprocal: a^(-n) = 1/a^n. Steps to solve: 1) Move the base to the denominator, 2) Make the exponent positive, 3) Simplify. For example, 2^(-3) = 1/2^3 = 1/8.

What are the most common mistakes students make with exponent rules?

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Common errors include: 1) Adding exponents when multiplying different bases (2^3 × 3^2 ≠ 5^5), 2) Multiplying exponents instead of adding when multiplying same bases, 3) Forgetting that only the exponents change, not the base, 4) Confusing negative exponents with negative numbers.

Can you use exponent rules with variables and algebraic expressions?

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Yes, exponent rules work the same way with variables. Examples: x^3 × x^5 = x^8, (y^2)^4 = y^8, and (2x)^3 = 2^3 × x^3 = 8x^3. The key is treating variables like any other base and applying the same rules consistently.

How do you handle exponents with fractions and multiplication?

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For fractional bases: (a/b)^n = a^n/b^n. For products: (a×b)^n = a^n × b^n. Example: (2×3)^2 = 2^2 × 3^2 = 4 × 9 = 36. These distributive properties make complex expressions easier to solve step by step.

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