How To Solve Algebraic Problems With Exponents: 8 Steps - WikiHow

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Terms of Use wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow How to Solve Algebraic Problems With Exponents PDF download Download Article Co-authored by David Jia

Last Updated: October 20, 2025 References

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  • Solving a Problem with Exponents
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  • Understanding the Laws of Exponents
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This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been viewed 83,840 times.

In algebra, the operations (adding, subtracting, multiplying, and dividing) performed on variables work the same as the operations performed on numbers. When performing these operations on exponents, however, the laws are different. By learning these special rules for exponents, you can easily simplify algebraic expressions that include them.

Steps

Method 1 Method 1 of 2:

Solving a Problem with Exponents

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  1. Step 1 Address the order of operations. 1 Address the order of operations. Just like any problem in mathematics, an algebraic problem must be completed by the order of operations. You can use the phrase "Please Excuse My Dear Aunt Sally," or the acronym PEMDAS, to help you remember Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.[1]
    • For example, if the problem is 4 ( x 10 ÷ x 4 ) ÷ 2 ( x 3 × x 2 ) − 3 x 0 {\displaystyle 4(x^{10}\div x^{4})\div 2(x^{3}\times x^{2})-3x^{0}} , you would first complete the calculations inside the parentheses.
  2. Step 2 Simplify the expressions using the laws of exponents. 2 Simplify the expressions using the laws of exponents. Remember, you can only simplify if the exponents have the same base.[2]
    • For example, x 10 ÷ x 4 {\displaystyle x^{10}\div x^{4}} can simplify to x 10 − 4 {\displaystyle x^{10-4}} , or x 6 {\displaystyle x^{6}} . x 3 × x 2 {\displaystyle x^{3}\times x^{2}} can simplify to x 3 + 2 {\displaystyle x^{3+2}} , or x 5 {\displaystyle x^{5}} . x 0 {\displaystyle x^{0}} is 1, since any number to the zero power is 1. So, the simplified problem becomes 4 ( x 6 ) ÷ 2 ( x 5 ) − 3 ( 1 ) {\displaystyle 4(x^{6})\div 2(x^{5})-3(1)} .
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  3. Step 3 Simplify coefficients. 3 Simplify coefficients. Coefficients are the numbers in an algebraic problem. When simplifying coefficients with exponents, you complete the regular operations.[3]
    • For example, for 4 ( x 6 ) ÷ 2 ( x 5 ) {\displaystyle 4(x^{6})\div 2(x^{5})} , you would first divide the coefficients: 4 ÷ 2 = 2 {\displaystyle 4\div 2=2} . Then, divide the exponents: x 6 ÷ x 5 {\displaystyle x^{6}\div x^{5}} = x 6 − 5 {\displaystyle x^{6-5}} = x 1 {\displaystyle x^{1}} = x {\displaystyle x} . Since 3 ( 1 ) {\displaystyle 3(1)} simplifies to 3 {\displaystyle 3} , the final, simplified problem is 2 x − 3 {\displaystyle 2x-3} .
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Method 2 Method 2 of 2:

Understanding the Laws of Exponents

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  1. Step 1 Solve expressions with a positive exponent. 1 Solve expressions with a positive exponent. An exponent simply tells you how many times you multiply the base (big number) by itself.[4]
    • For example, x 3 {\displaystyle x^{3}} is the same as x × x × x {\displaystyle x\times x\times x} .
    • Plugging in a number, you would have 2 3 {\displaystyle 2^{3}} = 2 × 2 × 2 {\displaystyle 2\times 2\times 2} = 8 {\displaystyle 8}
    • Expressions to the first degree (expressions with an exponent of 1) always simplify to the base. It is like saying “x one time.” For example, x 1 = x {\displaystyle x^{1}=x} .
    • Expressions to the zero degree (expressions with an exponent of 0) always simplify to 1. For example, x 0 = 1 {\displaystyle x^{0}=1} .
  2. Step 2 Simplify multiplication expressions with a positive exponent. 2 Simplify multiplication expressions with a positive exponent. When you multiply two exponents with the same base, you can simplify the expression by adding the exponents.[5] Do NOT add or multiply the base.[6]
    • This rule does not apply to numbers that have a different base. For example, you cannot simplify 2 3 × 3 2 {\displaystyle 2^{3}\times 3^{2}} , you simply have to solve the exponents separately and then multiply the two numbers.
    • For example, x 2 × x 4 {\displaystyle x^{2}\times x^{4}} is the same as x 2 + 4 {\displaystyle x^{2+4}} , which is the same as x 6 {\displaystyle x^{6}} .
    • Plugging in a number, you would have 2 2 × 2 4 {\displaystyle 2^{2}\times 2^{4}} = 2 2 + 4 {\displaystyle 2^{2+4}} = 2 6 {\displaystyle 2^{6}} = 2 × 2 × 2 × 2 × 2 × 2 {\displaystyle 2\times 2\times 2\times 2\times 2\times 2} = 64 {\displaystyle 64}
  3. Step 3 Simplify division expressions with a positive exponent. 3 Simplify division expressions with a positive exponent. When you divide to exponents with the same base, you can simplify the expression by subtracting the exponents.[7] Do NOT divide or subtract the base.
    • For example, x 10 x 5 {\displaystyle {\frac {x^{10}}{x^{5}}}} is the same as x 10 − 5 {\displaystyle x^{10-5}} , which is the same as x 5 {\displaystyle x^{5}} .
    • Plugging in a number, you would have 2 10 2 5 {\displaystyle {\frac {2^{10}}{2^{5}}}} = 2 10 − 5 {\displaystyle 2^{10-5}} = 2 5 {\displaystyle 2^{5}} = 2 × 2 × 2 × 2 × 2 {\displaystyle 2\times 2\times 2\times 2\times 2} = 32 {\displaystyle 32}
  4. Step 4 Simplify exponents with a positive exponent. 4 Simplify exponents with a positive exponent. Sometimes an exponent will have an exponent. In this situation, you would multiply the two exponents.[8]
    • For example, ( x 2 ) 3 {\displaystyle (x^{2})^{3}} is the same as x 2 × 3 {\displaystyle x^{2\times 3}} , which is the same as x 6 {\displaystyle x^{6}} .[9]
    • Plugging in a number, you would have ( 2 2 ) 3 {\displaystyle (2^{2})^{3}} = 2 2 × 3 {\displaystyle 2^{2\times 3}} = 2 6 {\displaystyle 2^{6}} = 2 × 2 × 2 × 2 × 2 × 2 {\displaystyle 2\times 2\times 2\times 2\times 2\times 2} = 64 {\displaystyle 64}
  5. Step 5 Simplify expressions with a negative exponent. 5 Simplify expressions with a negative exponent. You can think of a negative exponent as being the opposite of a positive exponent.[10] Since a positive exponent tells you how many times to multiply, a negative exponent tells you how many times to divide.[11] To simplify an expression with a negative exponent, use the formula x − a = 1 x a {\displaystyle x^{-a}={\frac {1}{x^{a}}}} .
    • For example, x − 4 {\displaystyle x^{-4}} is the same as 1 x 4 {\displaystyle {\frac {1}{x^{4}}}} .
    • Plugging in a number, 2 − 4 {\displaystyle 2^{-4}} = 1 2 4 {\displaystyle {\frac {1}{2^{4}}}} = 1 2 × 2 × 2 × 2 {\displaystyle {\frac {1}{2\times 2\times 2\times 2}}} = 1 16 {\displaystyle {\frac {1}{16}}}
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Community Q&A

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  • Question How can I solve an unknown exponent, e.g 2^y? Donagan Donagan Top Answerer The expression would have to be part of an equation before you could find the value of y. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 21 Helpful 13
  • Question How do I simplify algebraic expressions? Donagan Donagan Top Answerer See Simplify Algebraic Expressions. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 2 Helpful 6
  • Question How do I answer if the problem is in sentence form? Community Answer Community Answer Do your best to take key vocabulary and transform it into math, then take it from there. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 3 Helpful 6
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Video

Tips

  • An exponent only affects a variable that is to its immediate left, including signs. For example, − x 2 = − 1 ( x × x ) = − x {\displaystyle -x{^{2}}=-1(x\times x)=-x} , while ( − x ) 2 = − x × − x = x {\displaystyle (-x){^{2}}=-x\times -x=x} . Thanks Helpful 0 Not Helpful 0
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References

  1. ↑ https://www.mathsisfun.com/operation-order-pemdas.html
  2. ↑ https://content.nroc.org/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U11_L1_T2_text_final.html
  3. ↑ https://content.nroc.org/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U11_L1_T2_text_final.html
  4. ↑ https://www.mathsisfun.com/exponent.html
  5. ↑ David Jia. Academic Tutor. Expert Interview
  6. ↑ https://www.mathwarehouse.com/algebra/exponents/index.php
  7. ↑ https://www.mathwarehouse.com/algebra/exponents/laws-of-exponents.php
  8. ↑ https://www.mathwarehouse.com/algebra/exponents/laws-of-exponents.php
  9. ↑ David Jia. Academic Tutor. Expert Interview
More References (2)
  1. ↑ David Jia. Academic Tutor. Expert Interview
  2. ↑ http://www.mathsisfun.com/algebra/negative-exponents.html

About This Article

David Jia Co-authored by: David Jia Math Tutor This article was co-authored by David Jia. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 83,840 times. 11 votes - 62% Co-authors: 18 Updated: October 20, 2025 Views: 83,840 Categories: Exponents and Logarithms In other languagesSpanish
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