Question

What is the value of $x$ if $\frac{a^x}{a^y}=a^{10}$ and $(a^y{)}^3=a^x$ for $a>1$ ?

• A. $25$
• B. $5$
• C. $20$
• D. $15$

Answer 1

The correct option is D $15$

Lets rewrite the first expression,

$\frac{a^x}{a^y}=a^{10}$

$a^x=a^y\times a^{10}$

$a^x=a^{y+10}$

$⟹x=y+10$..........(1)

Similarly, rewriting the second expression,

$(a^y{)}^3=a^x$

$a^{3y}=a^x$

$⟹3y=x$..............(2)

From (1) & (2), we can write that that $3y=y+10$

$⟹y=5$

$⟹x=15$

Thus the answer is $x=15$

Note that we have used two concepts here,

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

2. $(x^a{)}^b=x^{a\times b}$