3.

(a) [tex]\( 5^8 \div 5^x = 5^2 \)[/tex]. Find the value of [tex]\( x \)[/tex]. [1]

(b) Simplify [tex]\( \left(x^5\right)^3 \)[/tex]. [1]



Answer :

Alright, let's solve the given problems step by step.

### Part (a)

We start with the equation:
[tex]\[ 5^8 \div 5^x = 5^2 \][/tex]

Using the properties of exponents, particularly the law of exponents that states:
[tex]\[ \frac{a^m}{a^n} = a^{m-n} \][/tex]

We can rewrite the left-hand side of the equation:
[tex]\[ 5^{8-x} = 5^2 \][/tex]

Since the bases on both sides of the equation are the same, we can set the exponents equal to each other:
[tex]\[ 8 - x = 2 \][/tex]

Solving for [tex]\( x \)[/tex]:
[tex]\[ 8 - x = 2 \][/tex]
[tex]\[ x = 8 - 2 \][/tex]
[tex]\[ x = 6 \][/tex]

So, the value of [tex]\( x \)[/tex] is:
[tex]\[ x = 6 \][/tex]

### Part (b)

We need to simplify the expression:
[tex]\[ \left( x^5 \right)^3 \][/tex]

Using the law of exponents that states:
[tex]\[ (a^m)^n = a^{m \cdot n} \][/tex]

We can simplify the expression:
[tex]\[ \left( x^5 \right)^3 = x^{5 \cdot 3} = x^{15} \][/tex]

So, the simplified expression is:
[tex]\[ x^{15} \][/tex]

### Summary

(a) The value of [tex]\( x \)[/tex] is [tex]\( 6 \)[/tex].

(b) The simplified form of [tex]\( \left( x^5 \right)^3 \)[/tex] is [tex]\( x^{15} \)[/tex].