Week 3 – Precalculus – Simplifying Expressions

Hello! This week I wanted to go over evaluating each expression and writing each answer as an integer or a fraction. This was something that I struggled with at the beginning, but after receiving clarification, I wanted to explain it to anyone else that may be struggling with it.

Say you wanted to evaluate the expression [( $\frac{5}{6}$ $)^{-3}$ $]^{\frac{2}{3}}$ .

Looks pretty intimidating, right? But I will walk you through the steps to evaluating this expression and by the time I finish explaining, this will be a whole lot simpler.

Let’s take a look at the fractional exponents. Consider the Power Rule of exponents ( ex. ( $(2^{2})^3$ — 2 and 3 would be multiplied to yield 6, which would make $2^{6}$ ) . If we apply the Power Rule of exponents to this expression, we should be able to evaluate this expression.

Simplify the two fractional exponents first. -3 is equivalent to $\frac{-3}{1}$ . This helps me when I am multiplying the fractional exponents together. When we’re multiplying fractions, remember to multiply across.

$\frac{-3}{1}$ multiplied by $\frac{2}{3}$ yields $\frac{-6}{3}$ .

Now, let’s take a look at this fractional exponent. Is there any way that we can simplify it further and make evaluating this expression easier for us? Well, by taking a look at $\frac{-6}{3}$ , we can realize that 3 can divide in to -6 twice, which gives us -2.

So, now we have ( $\frac{5}{2} )^-2$ . A negative exponent just means that the base is on the wrong side of the fraction line, so we want to flip the base to the other side to create a reciprocal, which turns that negative exponent in to a positive exponent. Now we have $\frac{6}{5} )^2$ .

Now that we have a positive exponent, our base is on the right side of the fraction line, we can apply the positive exponent to $\frac{6}{5}$ . This will give us $\frac{36}{25}$ , which is our final answer.

Let’s try another question.

[( $\frac{17}{25}$ $)^{-5}$ $]^{\frac{2}{5}}$ divided by [( $\frac{17}{25}$ $)^{6}$ $]^{\frac{-1}{3}}$ .

First, simplify the two fractional exponents on both sides. -5 is equivalent to $\frac{-5}{1}$ and 6 is equivalent to $\frac{6}{1}$ . Multiply across, $\frac{-5}{1}$ multiplied by $\frac{2}{5}$ and $\frac{6}{1}$ multiplied by $\frac{-1}{3}$ . The first pair will give us $\frac{-10}{5}$ , which we can simplify to -2, and the second pair will give us $\frac{-6}{3}$ which we can also simplify to 2.

Here, we can flip the bases to their reciprocal. Your equation should look like this: ( $\frac{25}{17} )^2$ divided by ( $\frac{25}{17} )^2$ .

Now, take a look at both sides. Notice they are the exact same. From here, we know that the expression will simplify down to 1 on both sides, and 1 divided by one will equal 1, which is our final answer.

I hope this helps!

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