Question

\frac{3x^{8}*7x^{3} }{3x^{6}*7 }" alt="\frac{3x^{8}*7x^{3} }{3x^{6}*7 }" align="absmiddle" class="latex-formula"> can be written in the form 3a⋅7b where: a= and b=

Answer 1

Answer:

$a = \dfrac{x^2}{3} \quad \textsf{and} \quad b = \dfrac{x^3}{7}$

Step-by-step explanation:

Given expression:

$\dfrac{3x^{8} \cdot 7x^{3} }{3x^{6} \cdot 7 }$

There are several ways this problem can be approached, and therefore many different answers.  The goal is to reduce the given expression to a simple product of two terms in x, set that to the given form 3a⋅7b then solve for a and b.

$\implies \dfrac{3x^{8} \cdot 7x^{3} }{3x^{6} \cdot 7}$

Cancel the common factors 3 and 7:

$\implies \dfrac{\diagup\!\!\!\!3x^{8} \cdot \diagup\!\!\!\!7x^{3} }{\diagup\!\!\!\!3x^{6} \cdot \diagup\!\!\!\!7}$

$\implies \dfrac{x^8 \cdot x^3}{x^6}$

Separate the fraction:

$\implies \dfrac{x^8}{x^6} \cdot x^3$

$\textsf{Apply the quotient rule of exponents} \quad \dfrac{a^b}{a^c}=a^{b-c}:$

$\implies x^{8-6} \cdot x^3$

$\implies x^{2} \cdot x^3$

Now equate the simplified expression to the given form:

$\implies x^{2} \cdot x^3=3a \cdot 7b$

Therefore:

$\begin{aligned}x^{2} &=3a \:\: &\textsf{ and }\:\: \quad x^3 &=7b\\ \Rightarrow a & = \dfrac{x^2}{3} & \Rightarrow b & = \dfrac{x^3}{7}\end{aligned}$

However, we could also separate them as:

$\begin{aligned}x^{3} &=3a \:\: &\textsf{ and }\:\: \quad x^2 &=7b\\ \Rightarrow a & = \dfrac{x^3}{3} & \Rightarrow b & = \dfrac{x^2}{7}\end{aligned}$

Another way of writing them would be to go back a few steps and separate the fraction in x terms differently:

$\implies \dfrac{x^8 \cdot x^3}{x^6}=x^8 \cdot \dfrac{x^3}{x^6}=x^8 \cdot x^{-3}$

Therefore, this would give us:

$\begin{aligned}x^{8} &=3a \:\: &\textsf{ and }\:\: \quad x^{-3} &=7b\\ \Rightarrow a & = \dfrac{x^8}{3} & \Rightarrow b & = \dfrac{1}{7x^3}\end{aligned}$

As the given expression reduces to x⁵, we can separate the x term in any way we like, so long as the coefficient of a is ¹/₃ and the coefficient of b is ¹/₇.  Therefore, there are many possible answers.

Answer 2

Answer:

$a=\frac{x^2}{3} \\\\ b= \frac{x^3}{7}$.

Step-by-step explanation:

1. Write the expression.

$\frac{3x^8*7x^3}{3x^6*7}$

2. Separate into 2 fractions.

$\frac{3x^8}{3x^6} *\frac{7x^3}{7}$

3. Divide the left hand side fraction by 3 and the right one by 7.

$\frac{3x^8}{3x^6*(3)} *\frac{7x^3}{7(7)}\\ \\\frac{3x^8}{9x^6} *\frac{7x^3}{49}$

4. Simplify.

$\frac{x^2}{3} *\frac{x^3}{7}$

5. Identify "a" and "b".

$a=\frac{x^2}{3} \\\\ b= \frac{x^3}{7}$

6. Rewrite in the form 3a⋅7b.

$3(\frac{x^2}{3})*7(\frac{x^3}{7})$

7. Verify.

If the re-writting was done correctly, then if we substitute x by a value both in the original expression and the re-written expression, it should give the same result. Let's test it with x= 2:

$\frac{3(2)^8*7(2)^3}{3(2)^6*7}=32\\\\3(\frac{(2)^2}{3})*7(\frac{(2)^3}{7})=32.$

8. Express your results.

$\frac{3x^8}{3x^6} *\frac{7x^3}{7}=3(\frac{x^2}{3})*7(\frac{x^3}{7})\\ \\a=\frac{x^2}{3} \\\\ b= \frac{x^3}{7}$