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How to Solve Exponents

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How to Solve Exponents

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Exponents are used when a number is multiplied by itself. You can also easily add, subtract, and multiply exponents for simplifying problems as needed when you have learned the property rules. Here is how to solve exponents. 

How to Solve Basic Exponents

Multiply the base repeatedly for the number of factors represented by the exponent.

–  82 = 8 × 8 = 64
–  53 = 5 × 5 × 5 = 125
– 45 = 4 × 4 × 4 × 4 × 4 = 1024

How to Add or Subtract Exponents with the Same Base and Exponent

You can only add or subtract exponents if they have the same identical bases and exponents. 

  • \mathbf{\2^{3} + 2^{3} = 1 *2^{3} + 1* 2^{3} = 2 * 2^{3} = 2 * 8 = 16}
  • \mathbf{3^{2}-3^{2}+5=5}
  • \mathbf{6x^{2}-3x^{2}=3x^{2}}

How to Solve Exponents with the Properties

  1. The product of powers property tells that when you multiply powers with the same base you just have to add the exponents.

    \boldsymbol{\mathbf{}x^{a} + x^{b} = x^{a+b}}

    For Example,  \mathbf{x^{2} + x^{3} = (x.x) + (x.x.x) = x^{5}}

  2.  The power of a power property says that to find a power of a power you just have to multiply the exponents.

    \mathbf{(x^{a})^{b}} = \mathbf{x^{ab}}

    For Example, \mathbf{(x^{2})^{5} = x^{2*5} = x^{10}}

  3. The power of a product property says when you raise a product to a power you raise each factor with a power. 

    \mathbf{(xy)^{a} = x^{a}y^{a} } 

    For Example, \mathbf{(2*3)^{2} = 2^{2}*3^{2} = 4 * 9 = 36}

  4. The quotient of powers property tells us that when you divide powers with the same base you just have to subtract the exponents.

    \mathbf{\frac{x^{a}}{x^{b}} = x^{a-b}, x \neq 0}

    For Example,  \mathbf{{\frac{3^{4}}{3^{2}}} = 3^{4-2} = 3^{2} = 9}

  5. Negative exponents are the reciprocals of the positive exponents.

    \mathbf{x^{-a} = \frac{1}{x^{a}},\ x \neq 0} \mathbf{x^{a} = \frac{1}{x^{-a}},\ x \neq 0}

    For Example, \mathbf{5^{-3} = \frac{1}{5^{3}}} , \mathbf{5^{3} = \frac{1}{5^{-3}}}

The same properties of exponents apply for both positive and negative exponents.

How to Solve Fractional Exponents

  1. Treat fractional exponents, like as a square root problem.

    \mathbf{\sqrt{x} = \sqrt[2]{x} = x^{\frac{1}{2}}}

  2. For mixed fractions, turn the top number into a normal exponent.

    For Example, \mathbf{x^{\frac{5}{2}} = x^{5} * x^{2} = (\sqrt{x})^{5}}

  3. Add, subtract, and multiply fractional exponents just like normal.

    For Example, \\ \mathbf{x^{\frac{7}{3}} + x^{\frac{7}{3}} = 2(x^{\frac{7}{3}}) }     \\ \mathbf{x^{\frac{7}{3}} * x^{\frac{2}{3}} = x^{\frac{9}{3}} = x^{3} }