Question

(SQRT5)^x=125.
I know it equals but I would love a step by step on how to solve

Answer 1

is this supposed to be
\[ \left( \sqrt{5}\right)^x = 125 \]

Answer 2

Yes

Answer 3

re-write the square root as 5 to the 1/ 2 power
\[ \left( 5^\frac{1}{2}\right)^x = 125 \]

also, it is useful to notice that 125 = 5*5*5 = 5^3
\[ \left( 5^\frac{1}{2}\right)^x = 5^3 \]

Answer 4

and we also want to use this rule
\[ \left( a^b\right)^c = a^{b\cdot c} \]

Answer 5

Why 5^1/2

Answer 6

notice that
\[ 5^\frac{1}{2} \cdot 5^\frac{1}{2} = 5^{\frac{1}{2} +\frac{1}{2} } =5^1 = 5\]

by definition, if you have A*A= 5 then A is the square root of 5
in other words,
\[ \sqrt{5} \text{ is the *same thing* as } 5^\frac{1}{2} \]

Answer 7

Ok

Answer 8

In general, the exponent version is easier to deal with.

Answer 9

can you apply the rule
\[ \left( a^b\right)^c = a^{b\cdot c} \]
to your problem
\[ \left( 5^\frac{1}{2}\right)^x = 5^3 \]?

Answer 10

you should get
\[ 5^\frac{x}{2} = 5^3 \]

the bases are equal , so you can equate the exponents:
\[ \frac{x}{2} = 3 \]
now solve for x by multiplying both sides by 2

Answer 11

6

Answer 12

yes, x=6 is the answer.

to review:

replace the square root with the exponent 1 /2
also, notice you have a base 5 on the left side. Try to make the right side also a base of 5
... in other words, expect you can write 125 as some power of 5.

then equate exponents.

Answer 13

Ok Thanks

Answer 14

I get it now