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One of the trickiest concepts in algebra involves the manipulation of exponents, or powers. (\(\frac{a}{b}\))\(^{0}\) = 1            = \(\frac{7 × 7 × 7 × 7}{7 × 7 × 7 × 7 × 7}\)                = 10²2\(^{-4}\) = \(\frac{1}{2^{4}}\) = \(\frac{1}{2}\) Ã— \(\frac{1}{2}\) × \(\frac{1}{2}\) × \(\frac{1}{2}\)  = \(\frac{1}{16}\)                         (2³)² = 2\(^{3 × 2}\)= 2⁶Here we will learn the Power of a Number. In symbols:\[\begin{aligned} \left(\dfrac{2 x^{5}}{y^{3}}\right)^{2}=\dfrac{\left(2 x^{5}\right)^{2}}{\left(y^{3}\right)^{2}} \quad \color {Red} \text {Raise numerator and denominator to the second power.}

The base a raised to the power of n is equal to the multiplication of a, n times: Examples: A. Many times, problems will require you to use the laws of exponents to simplify variables with exponents, or you will have to simplify an equation with exponents to solve it. White earned a Bachelor of Arts in history from Illinois Wesleyan University.We Have More Great Sciencing Articles!

Laws of Exponents: Let a be any number except 0 and m and n be two natural numbers.Then, First Law of Exponents: a m × a n = a m + n. Example 1: 3 2 × 3 3 = 3 2 + 3 = 3 5 = 243.

\end{aligned} \nonumber\]Simplify: \(\left(\dfrac{a^{4}}{3 b^{2}}\right)^{3}\)Raising a quotient to a power is similar to raising a product to a power.

When raising a product to a power, raise each of the factors to the indicated power.In each example we are raising a product to a power. Again, the key is to remember that the exponent tells us how many times to write the base as a factor, so we can write:Raising a number to the fourth power requires that we repeat that number as a factor four times (see Figure \(\PageIndex{1}\)).\[\begin{aligned} \left(a^{6} b^{4}\right)\left(a^{3} b^{2}\right) &= a^{6} b^{4} a^{3} b^{2} \quad \color {Red} \text { The associative property allows us to regroup in the order we prefer. } Many times, problems will require you to use the laws of exponents to simplify variables with exponents, or you will have to simplify an equation with exponents to solve it.

\\ &= x^{5 n-4-3+2 n} \quad \color {Red} \text { Distribute the minus sign. } Looking for math help for exponents? Multiplying powers with same base 1) If the bases are same and there is a multiplication between them then, add the exponents keeping the base common. B. C. 2. If \(n \neq 0\), then:\[\begin{aligned}\left(\frac{x}{y}\right)^{3} &=\frac{x}{y} \cdot \frac{x}{y} \cdot \frac{x}{y} \\ &=\frac{x \cdot x \cdot x}{y \cdot y \cdot y} \\ &=\frac{x^{3}}{y^{3}} \end{aligned} \nonumber \]This leads to the fifth and final law of exponents.\[\left(\dfrac{a}{b}\right)^{n}=\dfrac{a^{n}}{b^{n}} \nonumber \]\(\left(x^{2} y^{6}\right)\left(x^{4} y^{3}\right)\)However, it is much simpler to realize that when you raise a quotient to a power, you raise both numerator and denominator to that power. Exponentiation is not associative.For example, (2 3) 4 = 8 4 = 4096, whereas 2 (3 4) = 2 81 = 2 417 851 639 229 258 349 412 352.Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up (or left-associative).

Given \(a \neq 0\),To raise a quotient to a power, raise both numerator and denominator to that power.

The proof for this law is beyond the scope of this article.To divide two exponents that have the same base, subtract the power in the denominator from the power in the numerator.