1.4: Exponents
( \newcommand{\kernel}{\mathrm{null}\,}\)
The Laws of Exponents let you rewrite algebraic expressions that involve exponents. The last three listed here are really definitions rather than rules.
All variables here represent real numbers and all variables in denominators are nonzero.
- xa⋅xb=xa+b
- xaxb=xa−b
- (xa)b=xab
- (xy)a=xaya
- (xy)b=xbyb
- x0=1, provided x≠0. [Although in some contexts 00 is still defined to be 1.]
- x−n=1xn, provided x≠0.
- x1/n=n√x, provided x≠0.
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.
Rewrite 5x3 using negative exponents.
Solution
Since x−n=1xn, then x−3=1x3 and thus 5x3=5x−3.
Simplify (x−2y−3)2 as much as possible and write your answer using only positive exponents.
Solution
(x−2y−3)2=(x−2)2(y−3)2=x−4y−6=y6x4
Rewrite 4√x−3√x using exponents.
Solution
A square root is a radical with index of two. In other words, √x=2√x. Using the exponent rule above, √x=2√x=x1/2. Rewriting the square roots using the fractional exponent, 4√x−3√x=4x1/2−3x1/2.
Now we can use the negative exponent rule to rewrite the second term in the expression:
4x1/2−3x1/2=4x1/2−3x−1/2.
Rewrite (√p5)−1/3 using only positive exponents.
Solution
(√p5)−1/3=((p5)1/2)−1/3=p−5/6=1p5/6
Rewrite x−4/3as a radical.
Solution
x−4/3=1x4/3=1(x1/3)4(since 43=4⋅13)=1(3√x)4(using the radical equivalence)