Given the algebraic expression the quantity 27 times x to the five-thirds power end quantity to the negative one-third power comma, create an equivalent expression.

Question

A:  

B:  

C:

D:  

Answer 1

Hi David M

Could you include a picture of the Algebraic Expression, it reads like

(27x^5/3) is the quantity then the quantity is raised to negative 1/3 or -1/3

(27x^5/3)^-1/3

(27x^5/3)^-1/3 if this is your expression you could include parentheses in your description instead just end anyway for

(27x^5/3)^-1/3

Rewrite the expression as using the negative exponent rule

1/(27x^5/3)^1/3

Simplify the denominator

Apply the product rule

1/27^1/3(x^5/3)^1/3

Rewrite 27 as 3³

1/(3³)^1/3(x^5/3)^1/3

Apply the Power rule and multiple the exponents

1/(3¹)(x^5/9)

To give

1/3x^5/9

Please let me know if I’ve interpreted your Algebraic expression correctly, or simply include the expression. You can look up the rules for Exponents in your text or online. I hope this helps

Answer 2

The correct answer is A. This can be found by first taking 27 to the one-third power; that is, the cube root of 27. Since 3 x 3 x 3 = 27, the cube root of 27 is 3. Then, we find (x to the 5/3rd power) to the one-third power. Since powers raised to powers mean the exponents multiply ((6^4)^3 = 6^(4*3)), we get x to the 5/9th power. Finally, because we have the expression to the negative one-third power, we must take the reciprocal of the expression we get by multiplying the 3 by the x^(5/9), and when we do, we get the expression in A.

Answer 3

Hi David,

The worded expression, "27 times x to the 5/3 power end quantity to the negative one-third power" can be written as follows:

(27x^5/3)-1/3

^{From here, we have to use our exponent properties to simplify. Let's start outside of the parenthesis. The -1/3 will raise the ENTIRE expression to the -1/3 power, so we need to distribute it to each number.}

This would give us 27^-1/3 _x ^{5/3 (-1/3)}

(Note that when we raise a variable with an exponent to another power, we multiply both exponents.)

Now, we have 27^-1/3_x^-5/9

Both of these numbers are raised to a negative exponent. When we have a negative exponent, we put a "1" over the entire expression and make the exponent positive:

1

---------------

27^1/3_x^5/9

From here, we simplify: 27 ^1/3 _{means take the cubed root of 27, which is 3 (3x3x3=27). We can't simplify x}^{5/9 any further.}

Thus, our final expression is as follows:

1

--------------------

3x^5/9

^{I hope that helps!}