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:

\( 7^5\times2^3\times7^2\times2^4= \)

Examples with solutions for Exponents Rules

Step-by-step solutions included
Exercise #1

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 #2

Simplify the following equation:

53×24×52×23= 5^3\times2^4\times5^2\times2^3=

Step-by-Step Solution

Let's simplify the expression 53×24×52×23 5^3 \times 2^4 \times 5^2 \times 2^3 using the rules for exponents. We'll apply the product of powers rule, which states that when multiplying like bases, you can add the exponents.

  • Step 1: Focus on terms with the same base.
    Combine 53 5^3 and 52 5^2 . Since both terms have the base 55, we apply the rule am×an=am+na^m \times a^n = a^{m+n}: 53×52=53+2=55 5^3 \times 5^2 = 5^{3+2} = 5^5

  • Step 2: Combine 24 2^4 and 23 2^3 . Similarly, for the base 22: 24×23=24+3=27 2^4 \times 2^3 = 2^{4+3} = 2^7

After simplification, the expression becomes:
55×27 5^5 \times 2^7

Answer:

55×27 5^5\times2^7

Video Solution
Exercise #3

Reduce the following equation:

a2×a5×a3= a^2\times a^5\times a^3=

Step-by-Step Solution

To reduce the expression a2×a5×a3 a^2 \times a^5 \times a^3 , we will apply the product of powers property of exponents. This property states that when multiplying expressions with the same base, we add their exponents.

  • Step 1: Identify the exponents.
    The expression involves the same base a a with exponents: 2, 5, and 3.
  • Step 2: Add the exponents.
    According to the product of powers property, a2×a5×a3=a2+5+3 a^2 \times a^5 \times a^3 = a^{2+5+3} .
  • Step 3: Simplify the expression.
    Calculate the sum of the exponents: 2+5+3=10 2 + 5 + 3 = 10 . Therefore, the expression simplifies to a10 a^{10} .

Ultimately, the solution to the problem is a10 a^{10} . Among the provided choices, is correct: a10 a^{10} . The other options a5 a^5 , a8 a^8 , and a4 a^4 do not correctly reflect the sum of the exponents as calculated.

Answer:

a10 a^{10}

Video Solution
Exercise #4

Simplify the following equation:

64×23×62×25= 6^4\times2^3\times6^2\times2^5=

Step-by-Step Solution

To simplify the equation 64×23×62×25 6^4 \times 2^3 \times 6^2 \times 2^5 , we will make use of the rules of exponents, specifically the product of powers rule, which states that when multiplying two powers that have the same base, you can add their exponents.

Step 1: Identify and group the terms with the same base.
In the expression 64×23×62×25 6^4 \times 2^3 \times 6^2 \times 2^5 , group the powers of 6 together and the powers of 2 together:

  • Powers of 6: 64×62 6^4 \times 6^2

  • Powers of 2: 23×25 2^3 \times 2^5

Step 2: Apply the product of powers rule.
According to the product of powers rule, for any real number a a , and integers m m and n n , the expression am×an=am+n a^m \times a^n = a^{m+n} .

Apply this rule to the powers of 6:
64×62=64+2=66 6^4 \times 6^2 = 6^{4+2} = 6^6 .

Apply this rule to the powers of 2:
23×25=23+5=28 2^3 \times 2^5 = 2^{3+5} = 2^8 .

Step 3: Write down the final expression.
Combining our results gives the simplified expression: 66×28 6^6 \times 2^8 .

Therefore, the solution to the problem is 66×28 6^6 \times 2^8 .

Answer:

66×28 6^6\times2^8

Video Solution
Exercise #5

Insert the corresponding expression:

(x3)4= \left(x^3\right)^4=

Step-by-Step Solution

To simplify the expression (x3)4 (x^3)^4 , we'll follow these steps:

  • Step 1: Identify the expression: (x3)4 (x^3)^4 .
  • Step 2: Apply the formula for a power raised to another power.
  • Step 3: Calculate the product of the exponents.

Now, let's work through each step:

Step 1: We have the expression (x3)4 (x^3)^4 , which involves a power raised to another power.

Step 2: We apply the exponent rule (am)n=amn(a^m)^n = a^{m \cdot n} here with a=xa = x, m=3m = 3, and n=4n = 4.

Step 3: Multiply the exponents: 3×4=12 3 \times 4 = 12 . This gives us a new exponent for the base x x .

Therefore, (x3)4=x12(x^3)^4 = x^{12}.

Consequently, the correct answer choice is: x12 x^{12} from the options provided. The other options x6 x^6 , x1 x^1 , and x7 x^7 do not reflect the correct application of the exponent multiplication rule.

Answer:

x12 x^{12}

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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