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].
### 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].
