Algebra 1 help please.

Question

sqrt(14x^3)/sqrt(18x)

I'm supposed to simplify the expression and show or explain my work. I just don't understand how to simplify it, please help.

Answer 1

Hello John, thanks for your question.

This requires you to understand the rules for exponents.

sqrt(x) is the same thing as x^1/2. so, whatever you can do with exponents, you can do with square roots.

Here are the rules you need to solve this problem:

xⁿ/yⁿ = (x/y)ⁿ

(xy)ⁿ = xⁿyⁿ

xⁿ/x^m = x^n-m

(xⁿ)^m = x^nm

x¹ = x

Let's apply these rules to the problem you provided:

sqrt(14x³)/sqrt(18x), you can simplify the expression under one sqrt, because xⁿ/yⁿ = (x/y)ⁿ

sqrt(14x³/18x), you can simplify x³/x by thinking of it as x³/x¹ = x^3-1= x² (using the rule, xⁿ/x^m = x^n-m)

sqrt(14x²/18), reduce the fraction by removing the largest common factor, 2, from numerator and denominator. 14/18= 7/9

sqrt(7x²/9), you can distribute the sqrt to each term, because (xy)ⁿ = xⁿyⁿ and (x/y)ⁿ = xⁿ/yⁿ

sqrt(7) * sqrt(x²) / sqrt (9), evaluate the sqrts: sqrt x²= x, sqrt (9) = 3, the sqrt (7) = sqrt (7)

sqrt(7) * x /3, this is the simplified expression.

A few things to keep in mind when it comes to working with exponents:

(x +y)ⁿ does NOT equal xⁿ + yⁿ

Also, some numbers can be partially rooted if a factor of that number is a perfect square. For example, sqrt (27). Factors of 27 are 1, 3, 9, and 27. 9 is perfect square (3*3 =9). You can think of sqrt(27) as sqrt (9*3) which is equal to sqrt (9)* sqrt(3) which equals 3*sqrt(3). So, sqrt (27)= 3sqrt(3).

And lastly, the reason why sqrt(x²) is x is because of the rule (xⁿ)^m = x^nm. sqrt(x²) is the same thing as (x²)^1/2 = x^2*1/2 = x¹= x

I hope this helps!

Answer 2

When you have sq rt A / sq rt B, you can combine as sq rt (A/B).

So sq rt 14x³/sq rt 18x = sq rt (14x³/18x) = sq rt (7x²/9) = (x/3) sq rt 7