Combined Exponent Rules Practice Problems & Solutions

Master all exponent properties together with step-by-step practice problems. Learn to apply multiplication, division, power of power, and negative exponent rules in complex expressions.

📚Master Complex Exponent Problems with Combined Rules
  • Apply multiple exponent properties in single expressions with confidence
  • Simplify complex expressions involving parentheses and negative exponents
  • Convert negative exponents to positive form using proper techniques
  • Combine fractions with different bases using exponent rules
  • Solve multi-step problems requiring power of power and quotient rules
  • Handle expressions with variables in both base and exponent positions

Understanding Applying Combined Exponents Rules

Complete explanation with examples

Taking advantage of all the properties of powers or laws of exponents

From time to time, we will come across exercises in which we must use all the properties of powers together.
As soon as you have the exercise, try to first get rid of the parentheses according to the properties of powers and then, apply these properties to the corresponding terms, one after the other.

All the properties of powers or laws of exponents are:
am×an=a(m+n)a^m\times a^n=a^{(m+n)}
aman=a(m−n)\frac {a^m}{a^n} =a^{(m-n)}
(a×b)n=an×bn(a\times b)^n=a^n\times b^n
(ab)n=anbn(\frac {a}{b})^n=\frac {a^n}{b^n}
(an)m=a(n∗m)(a^n )^m=a^{(n*m)}
a0=1a^0=1
When a≠0a≠0
a−n=1ana^{-n}=\frac {1}{a^n}

Detailed explanation

Practice Applying Combined Exponents Rules

Test your knowledge with 44 quizzes

Simplify the following equation:

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

Examples with solutions for Applying Combined Exponents Rules

Step-by-step solutions included
Exercise #1

2423= \frac{2^4}{2^3}=

Step-by-Step Solution

Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

bmbn=bm−n \frac{b^m}{b^n}=b^{m-n}

We apply it in the problem:

2423=24−3=21 \frac{2^4}{2^3}=2^{4-3}=2^1

Remember that any number raised to the 1st power is equal to the number itself, meaning that:

b1=b b^1=b

Therefore, in the problem we obtain:

21=2 2^1=2

Therefore, the correct answer is option a.

Answer:

2 2

Video Solution
Exercise #2

9993= \frac{9^9}{9^3}=

Step-by-Step Solution

Note that in the fraction and its denominator, there are terms with the same base, so we will use the law of exponents for division between terms with the same base:

bmbn=bm−n \frac{b^m}{b^n}=b^{m-n}

Let's apply it to the problem:

9993=99−3=96 \frac{9^9}{9^3}=9^{9-3}=9^6

Therefore, the correct answer is b.

Answer:

96 9^6

Video Solution
Exercise #3

(35)4= (3^5)^4=

Step-by-Step Solution

To solve the exercise we use the power property:(an)m=an⋅m (a^n)^m=a^{n\cdot m}

We use the property with our exercise and solve:

(35)4=35×4=320 (3^5)^4=3^{5\times4}=3^{20}

Answer:

320 3^{20}

Video Solution
Exercise #4

(62)13= (6^2)^{13}=

Step-by-Step Solution

We use the formula:

(an)m=an×m (a^n)^m=a^{n\times m}

Therefore, we obtain:

62×13=626 6^{2\times13}=6^{26}

Answer:

626 6^{26}

Video Solution
Exercise #5

Solve the exercise:

(a5)7= (a^5)^7=

Step-by-Step Solution

We use the formula:

(am)n=am×n (a^m)^n=a^{m\times n}

and therefore we obtain:

(a5)7=a5×7=a35 (a^5)^7=a^{5\times7}=a^{35}

Answer:

a35 a^{35}

Video Solution

Frequently Asked Questions

Everything you need to know about Applying Combined Exponents Rules

What order should I use when applying multiple exponent rules?

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Start with parentheses first, then apply power of power rules, followed by multiplication and division of same bases. Finally, convert negative exponents to positive form if required.

How do I handle expressions like (3/x)^-3 × (x^-2)^4?

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First apply the quotient power rule to (3/x)^-3, then use power of power for (x^-2)^4. Combine like bases using multiplication and division rules, then convert negative exponents to fractions.

What are all the exponent properties I need to know?

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The seven key properties are: 1) a^m × a^n = a^(m+n), 2) a^m ÷ a^n = a^(m-n), 3) (ab)^n = a^n × b^n, 4) (a/b)^n = a^n/b^n, 5) (a^m)^n = a^(mn), 6) a^0 = 1, 7) a^-n = 1/a^n.

Why do I get confused with negative exponents in complex problems?

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Negative exponents create reciprocals, which can make expressions look complicated. Practice converting a^-n = 1/a^n early in your solution, and remember that negative times negative equals positive when subtracting exponents.

How do I simplify (x^2)^-3 × x^5 ÷ x^-2?

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Step by step: (x^2)^-3 becomes x^-6, then x^-6 × x^5 = x^-1, finally x^-1 ÷ x^-2 = x^-1-(-2) = x^1 = x.

What's the difference between (2x)^3 and 2x^3?

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(2x)^3 = 2^3 × x^3 = 8x^3 because the parentheses mean both 2 and x are cubed. However, 2x^3 means only x is cubed, so it equals 2 × x^3.

Can I use exponent rules when bases are different?

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You can only combine exponents when bases are identical. For different bases like 2^3 × 3^3, you cannot simplify further unless you can rewrite one base in terms of the other (like 4^2 = (2^2)^2 = 2^4).

How do I check if my exponent simplification is correct?

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Substitute simple numbers for variables and calculate both the original expression and your simplified answer. If they match, your simplification is likely correct. Also verify that you've applied each rule properly in sequence.

More Applying Combined Exponents Rules Questions

Click on any question to see the complete solution with step-by-step explanations

Applying Combined Exponents Rules

Calculate (2/3)³: Finding the Cube of a Fraction Solve: X³⋅X²÷X⁵+X⁴ - Simplifying Complex Exponent Operations Simplify the Expression: Y² + Y⁶ - Y⁵Y Polynomial Problem Solve: a²÷a + a³×a⁵ Expression Using Laws of Exponents Simplify the Expression: 5^-3 × 5^0 × 5^2 × 5^5 Using Laws of Exponents

Continue Your Math Journey

Master Applying Combined Exponents Rules first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

Multiplying Exponents with the Same BaseDivision of Exponents with the Same BaseExponent of a MultiplicationPower of a QuotientPower of a Power

Topics Learned in Later Sections

Rules of ExponentiationCombining Powers and Roots

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems
Applying the formula Calculating powers with negative exponents Complete the equation converting Negative Exponents to Positive Exponents Factoring Out the Greatest Common Factor (GCF) Factorization Identify the greater value More than one unknown Multiplying Exponents with the same base Number of terms A power law Presenting powers in the denominator as powers with negative exponents Presenting powers with negative exponents as fractions Presenting powers with negative exponents as fractions Monomial Single Variable Trinomial Binomial Two Variables Using laws of exponents with parameters Using multiple rules Using the laws of exponents Using variables Variable in the base of the power Variable in the exponent of the power Variables in the exponent of the power Worded problems The power of zero