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1.4: Exponents

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The Laws of Exponents let you rewrite algebraic expressions that involve exponents. The last three listed here are really definitions rather than rules.

Laws of Exponents

All variables here represent real numbers and all variables in denominators are nonzero.

  1. xaxb=xa+b
  2. xaxb=xab
  3. (xa)b=xab
  4. (xy)a=xaya
  5. (xy)b=xbyb
  6. x0=1, provided x0. [Although in some contexts 00 is still defined to be 1.]
  7. xn=1xn, provided x0.
  8. x1/n=nx, provided x0.

Example 1.4.1

Simplify (2x2)3(4x).

Solution

We'll begin by simplifying the (2x2)3 portion. Using Property 4, we can write

23(x2)3(4x)  
8x6(4x) Evaluate 23, and use Property 3.
32x7 Multiply the constants, and use Property 1, recalling x=x1.

Being able to work with negative and fractional exponents will be very important later in this course.

Example 1.4.2

Rewrite 5x3 using negative exponents.

Solution

Since xn=1xn, then x3=1x3 and thus 5x3=5x3.

Example 1.4.3

Simplify (x2y3)2 as much as possible and write your answer using only positive exponents.

Solution

(x2y3)2=(x2)2(y3)2=x4y6=y6x4

Example 1.4.4

Rewrite 4x3x using exponents.

Solution

A square root is a radical with index of two. In other words, x=2x. Using the exponent rule above, x=2x=x1/2. Rewriting the square roots using the fractional exponent, 4x3x=4x1/23x1/2.

Now we can use the negative exponent rule to rewrite the second term in the expression:

4x1/23x1/2=4x1/23x1/2.

Example 1.4.5

Rewrite (p5)1/3 using only positive exponents.

Solution

(p5)1/3=((p5)1/2)1/3=p5/6=1p5/6

Example 1.4.6

Rewrite x4/3as a radical.

Solution

x4/3=1x4/3=1(x1/3)4(since 43=413)=1(3x)4(using the radical equivalence)


This page titled 1.4: Exponents is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Shana Calaway, Dale Hoffman, & David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.

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