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Course: Algebra 2>Unit 6

Lesson 5: Solving exponential equations using properties of exponents

What are rational exponents?

Rational exponents are just like regular exponents, except the exponent is a fraction instead of a whole number. For example, ${2}^{{}^{\frac{3}{4}}}$ has a rational exponent, while ${2}^{3}$ has a whole number exponent.

Where are rational exponents and radicals used in the real world?

Rational exponents and radicals show up in a lot of places! For example, they're used in physics to calculate things like electrical resistance and wave frequencies. They're also used in finance to calculate compound interest.

What are some properties of exponents with rational exponents?

Here are some properties of rational exponents—notice how they are similar the properties of integer exponents:
• ${b}^{{}^{\frac{m}{n}}}=\sqrt[n]{\phantom{A}{b}^{m}}$
• ${b}^{{}^{\frac{m}{n}}}×{b}^{{}^{\frac{p}{n}}}={b}^{{}^{\frac{m+p}{n}}}$
• ${\left({b}^{{}^{\frac{m}{n}}}\right)}^{\frac{p}{q}}={b}^{{}^{\frac{mp}{nq}}}$

How do we evaluate exponents and radicals?

To evaluate an exponential expression, you can either use the properties of exponents to simplify it, or use a calculator. For radicals, you can find the square root, cube root, or ${n}^{\text{th}}$ root of a number using a calculator or by breaking the number down into its factors.

How do I convert between different forms of exponential expressions?

There are a few different ways you can write an exponential expression, and sometimes we might need to convert between them. Here are three equivalent forms for the same expression:
• ${2}^{{}^{\frac{3}{4}}}$
• $\sqrt[4]{\phantom{A}{2}^{3}}$
• $\sqrt[4]{\phantom{A}8}$

How can we solve exponential equations using properties of exponents?

To solve exponential equations, we can often use properties of exponents to simplify them. For example, to solve ${2}^{x}=8$, we can rewrite $8$ as ${2}^{3}$, which lets us set $x=3$.

Want to join the conversation?

• great practice, I can now work with exponents pretty well and before it was kind of intimidating but now its just challenging and fun to try to find and use the correct exponent properties and there's a lot of small stuff that if you miss you will get the answer wrong so you have to be precise and it keeps you on your toes (mentally) haha.
• I have many question about the meaning of life.
• There is no meaning of life, you have to make your own meaning to life.
• I have no questions
• great!
• i have no questions
• for the first property of exponents in this article, can the m be outside of the root or does it have to be under the root?
• Yes, good observation.
n√(b^m) = (n√b)^m

To show this, lets first simplify L.H.S.
n√(b^m)
= b^(m * 1 / n)
= b^(m / n)

R.H.S.
(n√b)^m
= (b^(1 / n))^m
= b^(1 / n * m)
= b^(m / n)