How to Solve Algebraic Problems With Exponents
Solve expressions with a positive exponent., Simplify multiplication expressions with a positive exponent., Simplify division expressions with a positive exponent., Simplify exponents with a positive exponent., Simplify expressions with a negative...
Step-by-Step Guide
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Step 1: Solve expressions with a positive exponent.
An exponent simply tells you how many times you multiply the base (big number) by itself.
For example, x3{\displaystyle x^{3}} is the same as x×x×x{\displaystyle x\times x\times x}.
Plugging in a number, you would have23{\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, x1=x{\displaystyle x^{1}=x}.
Expressions to the zero degree (expressions with an exponent of 0) always simplify to
1.
For example, x0=1{\displaystyle x^{0}=1}. -
Step 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.
Do NOT add or multiply the base.
This rule does not apply to numbers that have a different base.
For example, you cannot simplify 23×32{\displaystyle 2^{3}\times 3^{2}}, you simply have to solve the exponents separately and then multiply the two numbers.
For example, x2×x4{\displaystyle x^{2}\times x^{4}} is the same as x2+4{\displaystyle x^{2+4}}, which is the same as x6{\displaystyle x^{6}}.
Plugging in a number, you would have22×24{\displaystyle 2^{2}\times 2^{4}}=22+4{\displaystyle 2^{2+4}}=26{\displaystyle 2^{6}}=2×2×2×2×2×2{\displaystyle 2\times 2\times 2\times 2\times 2\times 2}=64{\displaystyle 64} , When you divide to exponents with the same base, you can simplify the expression by subtracting the exponents.Do NOT divide or subtract the base.
For example, x10x5{\displaystyle {\frac {x^{10}}{x^{5}}}} is the same as x10−5{\displaystyle x^{10-5}}, which is the same as x5{\displaystyle x^{5}}.
Plugging in a number, you would have 21025{\displaystyle {\frac {2^{10}}{2^{5}}}}=210−5{\displaystyle 2^{10-5}}=25{\displaystyle 2^{5}}=2×2×2×2×2{\displaystyle 2\times 2\times 2\times 2\times 2}=32{\displaystyle 32} , Sometimes an exponent will have an exponent.
In this situation, you would multiply the two exponents.For example, (x2)3{\displaystyle (x^{2})^{3}} is the same as x2×3{\displaystyle x^{2\times 3}}, which is the same as x6{\displaystyle x^{6}}.
Plugging in a number, you would have(22)3{\displaystyle (2^{2})^{3}}=22×3{\displaystyle 2^{2\times 3}}=26{\displaystyle 2^{6}}=2×2×2×2×2×2{\displaystyle 2\times 2\times 2\times 2\times 2\times 2}=64{\displaystyle 64} , You can think of a negative exponent as being the opposite of a positive exponent.
Since a positive exponent tells you how many times to multiply, a negative exponent tells you how many times to divide.To simplify an expression with a negative exponent, use the formula x−a=1xa{\displaystyle x^{-a}={\frac {1}{x^{a}}}}.
For example, x−4{\displaystyle x^{-4}} is the same as 1x4{\displaystyle {\frac {1}{x^{4}}}}.
Plugging in a number,2−4{\displaystyle 2^{-4}}=124{\displaystyle {\frac {1}{2^{4}}}}=12×2×2×2{\displaystyle {\frac {1}{2\times 2\times 2\times 2}}}=116{\displaystyle {\frac {1}{16}}} , 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.For example, if the problem is 4(x10÷x4)÷2(x3×x2)−3x0{\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. , Remember, you can only simplify if the exponents have the same base.
For example, x10÷x4{\displaystyle x^{10}\div x^{4}} can simplify to x10−4{\displaystyle x^{10-4}}, or x6{\displaystyle x^{6}}. x3×x2{\displaystyle x^{3}\times x^{2}} can simplify to x3+2{\displaystyle x^{3+2}}, or x5{\displaystyle x^{5}}.x0{\displaystyle x^{0}} is 1, since any number to the zero power is
1.
So, the simplified problem becomes 4(x6)÷2(x5)−3(1){\displaystyle 4(x^{6})\div 2(x^{5})-3(1)}. , Coefficients are the numbers in an algebraic problem.
When simplifying coefficients with exponents, you complete the regular operations.
For example, for 4(x6)÷2(x5){\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:x6÷x5{\displaystyle x^{6}\div x^{5}}=x6−5{\displaystyle x^{6-5}}=x1{\displaystyle x^{1}}=x{\displaystyle x}.
Since 3(1){\displaystyle 3(1)} simplifies to 3{\displaystyle 3}, the final, simplified problem is 2x−3{\displaystyle 2x-3}. -
Step 3: Simplify division expressions with a positive exponent.
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Step 4: Simplify exponents with a positive exponent.
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Step 5: Simplify expressions with a negative exponent.
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Step 6: Address the order of operations.
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Step 7: Simplify the expressions using the laws of exponents.
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Step 8: Simplify coefficients.
Detailed Guide
An exponent simply tells you how many times you multiply the base (big number) by itself.
For example, x3{\displaystyle x^{3}} is the same as x×x×x{\displaystyle x\times x\times x}.
Plugging in a number, you would have23{\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, x1=x{\displaystyle x^{1}=x}.
Expressions to the zero degree (expressions with an exponent of 0) always simplify to
1.
For example, x0=1{\displaystyle x^{0}=1}.
When you multiply two exponents with the same base, you can simplify the expression by adding the exponents.
Do NOT add or multiply the base.
This rule does not apply to numbers that have a different base.
For example, you cannot simplify 23×32{\displaystyle 2^{3}\times 3^{2}}, you simply have to solve the exponents separately and then multiply the two numbers.
For example, x2×x4{\displaystyle x^{2}\times x^{4}} is the same as x2+4{\displaystyle x^{2+4}}, which is the same as x6{\displaystyle x^{6}}.
Plugging in a number, you would have22×24{\displaystyle 2^{2}\times 2^{4}}=22+4{\displaystyle 2^{2+4}}=26{\displaystyle 2^{6}}=2×2×2×2×2×2{\displaystyle 2\times 2\times 2\times 2\times 2\times 2}=64{\displaystyle 64} , When you divide to exponents with the same base, you can simplify the expression by subtracting the exponents.Do NOT divide or subtract the base.
For example, x10x5{\displaystyle {\frac {x^{10}}{x^{5}}}} is the same as x10−5{\displaystyle x^{10-5}}, which is the same as x5{\displaystyle x^{5}}.
Plugging in a number, you would have 21025{\displaystyle {\frac {2^{10}}{2^{5}}}}=210−5{\displaystyle 2^{10-5}}=25{\displaystyle 2^{5}}=2×2×2×2×2{\displaystyle 2\times 2\times 2\times 2\times 2}=32{\displaystyle 32} , Sometimes an exponent will have an exponent.
In this situation, you would multiply the two exponents.For example, (x2)3{\displaystyle (x^{2})^{3}} is the same as x2×3{\displaystyle x^{2\times 3}}, which is the same as x6{\displaystyle x^{6}}.
Plugging in a number, you would have(22)3{\displaystyle (2^{2})^{3}}=22×3{\displaystyle 2^{2\times 3}}=26{\displaystyle 2^{6}}=2×2×2×2×2×2{\displaystyle 2\times 2\times 2\times 2\times 2\times 2}=64{\displaystyle 64} , You can think of a negative exponent as being the opposite of a positive exponent.
Since a positive exponent tells you how many times to multiply, a negative exponent tells you how many times to divide.To simplify an expression with a negative exponent, use the formula x−a=1xa{\displaystyle x^{-a}={\frac {1}{x^{a}}}}.
For example, x−4{\displaystyle x^{-4}} is the same as 1x4{\displaystyle {\frac {1}{x^{4}}}}.
Plugging in a number,2−4{\displaystyle 2^{-4}}=124{\displaystyle {\frac {1}{2^{4}}}}=12×2×2×2{\displaystyle {\frac {1}{2\times 2\times 2\times 2}}}=116{\displaystyle {\frac {1}{16}}} , 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.For example, if the problem is 4(x10÷x4)÷2(x3×x2)−3x0{\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. , Remember, you can only simplify if the exponents have the same base.
For example, x10÷x4{\displaystyle x^{10}\div x^{4}} can simplify to x10−4{\displaystyle x^{10-4}}, or x6{\displaystyle x^{6}}. x3×x2{\displaystyle x^{3}\times x^{2}} can simplify to x3+2{\displaystyle x^{3+2}}, or x5{\displaystyle x^{5}}.x0{\displaystyle x^{0}} is 1, since any number to the zero power is
1.
So, the simplified problem becomes 4(x6)÷2(x5)−3(1){\displaystyle 4(x^{6})\div 2(x^{5})-3(1)}. , Coefficients are the numbers in an algebraic problem.
When simplifying coefficients with exponents, you complete the regular operations.
For example, for 4(x6)÷2(x5){\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:x6÷x5{\displaystyle x^{6}\div x^{5}}=x6−5{\displaystyle x^{6-5}}=x1{\displaystyle x^{1}}=x{\displaystyle x}.
Since 3(1){\displaystyle 3(1)} simplifies to 3{\displaystyle 3}, the final, simplified problem is 2x−3{\displaystyle 2x-3}.
About the Author
Jason Collins
Experienced content creator specializing in lifestyle guides and tutorials.
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