Question Video: Computing Numerical Expressions Involving Square Roots Using Laws of Exponents | Nagwa Question Video: Computing Numerical Expressions Involving Square Roots Using Laws of Exponents | Nagwa

# Question Video: Computing Numerical Expressions Involving Square Roots Using Laws of Exponents Mathematics • Second Year of Preparatory School

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Simplify ((√3)² × (−√3)¹³)/(√3)³.

04:17

### Video Transcript

Simplify the square root of three squared multiplied by the negative square root of three to the power of 13 over the square root of three cubed.

The first thing we notice is how each parentheses contains the square root of three, with the exception of the second parentheses in the numerator, which contains the negative square root of three. We could use properties of exponents to factor the negative one out of the second parentheses, specifically using the power of a product rule. This rule states that the product of 𝑎 and 𝑏 to the power of 𝑛 is equal to the product of 𝑎 to the power of 𝑛 times 𝑏 to the power of 𝑛 for nonzero real numbers 𝑎 and 𝑏 and integer 𝑛.

We can write the negative square root of three to the power of 13 as negative one times the square root of three to the power of 13, then, following this rule, negative one to the power of 13 times the square root of three to the power of 13. So the numerator of our expression is now the square root of three squared times negative one to the power of 13 times the square root of three to the power of 13. The denominator has not been changed.

Next, we will evaluate negative one to the power of 13. We know that negative one squared equals positive one, negative one cubed equals negative one, negative one to the power of four equals one, negative one to the power of five equals negative one, and so on. As the pattern reveals, any even power of negative one equals one, and any odd power of negative one equals negative one. Therefore, negative one to the power of 13 equals negative one.

Because multiplication is commutative and associative, we can move the factor of negative one to the front of the rational expression. Next, we will simplify the numerator using the product rule for exponents. We recall that this rule states that 𝑎 to the power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 plus 𝑛, where 𝑎 is a nonzero real number and 𝑚 and 𝑛 are integers.

Using the product rule, we can write the numerator as the square root of three to the power of two plus 13. So we now have negative one times the quotient of the square root of three to the power of 15 over the square root of three cubed. Now we recall the quotient rule for exponents, which states that the quotient of 𝑎 to the power of 𝑚 over 𝑎 to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 minus 𝑛, where 𝑎 is a nonzero real number and 𝑚 and 𝑛 are integers. This means we have the square root of three to the power of 15 minus three. So the new power of the square root of three is 12.

Now in order to simplify the square root of three to the power of 12, we will use the product of squares along with the following rule. For 𝑎 greater than zero, the square root of 𝑎 squared is equal to 𝑎. So next, we use an expanded form of the product rule in reverse to write out the square root of three to the power of 12 as the product of six factors of the square root of three squared. Each square root of three squared equals three. In short, we can now write the expression as negative one times three to the power of six. And three to the power of six is 729. Finally, we multiply negative one and 729.

In conclusion, we have shown that the square root of three squared multiplied by the negative square root of three to the power of 13 over the square root of three cubed simplifies to negative 729.

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