Pre-Algebra - Product Rule of Exponents

Simplify:

This expression is already simplified

$(xy)^{12}$

Explanation

The product rule tells us that if we're multiplying 2 terms with exponents, we add the exponents. However, that is only applicable for 2 terms with the same base. In this case, our bases are different variables, so we can't use the product rule. This expression is as simplified as possible.

Simplify:

$\left (5t^{3} \right )^{ 3}$

$125 t^{9}$

$125 t^{6}$

$1 5 t^{9}$

$1 5 t^{6}$

$125 t^{3}$

Explanation

$\left (5t^{3} \right )^{ 3} = 5^{ 3} \cdot \left (t^{3} \right ) ^{ 3} = 125 t^{3 \cdot 3} = 125 t^{9}$

Simplify the following expression:

Explanation

The exponent represents how many times the term is being multiplied. So, for example, means and would be

So the first term

And the second term

Since the two terms are only separated by parentheses, they are being all multiplied together.

First multiply the coefficients,

We also have a total of 6 's and 1 all being multipled together.

The final answer is

Which of the following is equal to $3x^2\times 4x^3$ ?

$\small 12x^5$

` $\small 12x^6$

$\small 6x^6$

$\small 6x^5$

$\small 7x^5$

Explanation

Remember that when multiplying variables with exponents, the following property holds true:

$x^a\times x^b = x^{a+b}$

With this knowledge, we can solve the problem:

$3x^2\times 4x^3$

$=(3\times 4)x^2^+^3$

$\small =12x^5$

The answer is $\small 12x^5$ .

Simplify the following:

$x^{10}*x^2$

$x^{12}$

$x^{20}$

None of the above

Explanation

When you multiply exponents you will end up adding the two numbers so you will get:

$x^{10+2}$ which is $x^{12}$

Simplify:

$y^{14}$

$y^{45}$

$y^{-4}$

Explanation

When you multiply exponents with the same base, you add the exponents:

$(y^5)(y^9)=y^{14}$

Simplify:

$x^{3}y^{2}\cdot xy^{4}$

$x^{4}y^{6}$

$x^{3}y^{8}$

$x^{3}y^{6}$

$x^{4}y^{8}$

Explanation

Use the product rule of exponents, which states that when multiplying expression(s) with the same base, you add the exponents together:

$x^{3+1}y^{4+2}$

Therefore the correct answer is $x^{4}y^{6}$ .

Simplify:

$3x^{2}2y^{7}\times 5x^{5}y^{3}$

$30x^{7}y^{10}$

$15x^{7}2y^{10}$

$30xy^{17}$

$30x^{17}y$

$30x^{10}y^{21}$

Explanation

When multiplying variables with exponents, we must remember the Product Rule of Exponents: $x^{m}*x^{n}=x^{m+n}$

Step 1: Reorganize the terms so the terms are together:

$3x^{2}2y^{7}\times 5x^{5}y^{3} = 3\times2\times5\times x^{2} \times x^{5}\times y^{7} \times y^{3}$

Step 2: Multiply $3\times2\times5$ :

$30\times x^{2}\times x^{5}\times y^{7}\times y^{3}$

Step 3: Use the Product Rule of Exponents to combine $x^{2}$ and $x^{5}$ , and then _ $y^{3}$ _and $y^{7}$ :

$30\times x^{7}\times y^{10}$

$30x^{7}y^{10}$

Simplify:

Explanation

To simplify it helps to understand what it means. just means x times itself 3 times, . means multiply times itself, or . We can simplify that by multiplying and . Multiplying both terms together gives us . This is a 9 and 5 x's being multiplied together, or .

Simplify the following expression:

Explanation

Remember the rule:

"When you MULTIPLY terms together, simplify by ADDING the exponents of each variable."

Here's what that looks like in this case:

First multiply the coefficients:

Then ADD the exponents of the variables to simplify. In the first term, the exponent on the is 2. In the second term the exponent is 1. So we ADD and just have .