Answer :
Certainly! Let's rewrite the given exponents using relevant exponent rules step by step.
### a. Rewrite [tex]\(4^{5x}\)[/tex] using exponent rules
First, recognize that [tex]\(4\)[/tex] can be expressed as [tex]\(2^2\)[/tex]:
[tex]\[ 4 = 2^2 \][/tex]
So, [tex]\(4^{5x}\)[/tex] becomes:
[tex]\[ 4^{5x} = (2^2)^{5x} \][/tex]
Using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we can simplify:
[tex]\[ (2^2)^{5x} = 2^{2 \cdot 5x} = 2^{10x} \][/tex]
Thus:
[tex]\[ 4^{5x} \can be rewritten as \boxed{2^{10x}} \][/tex]
### b. Rewrite [tex]\(3^{3x+1}\)[/tex] using exponent rules
We can use the property of exponents that states [tex]\(a^{m+n} = a^m \cdot a^n\)[/tex]:
[tex]\[ 3^{3x+1} = 3^{3x} \cdot 3^1 \][/tex]
Thus:
[tex]\[ 3^{3x+1}\ can be rewritten as \boxed{3^{3x} \cdot 3^1} \][/tex]
### c. Rewrite [tex]\(8^{x-y}\)[/tex] using exponent rules
First, recognize that [tex]\(8\)[/tex] can be expressed as [tex]\(2^3\)[/tex]:
[tex]\[ 8 = 2^3 \][/tex]
So, [tex]\(8^{x-y}\)[/tex] becomes:
[tex]\[ 8^{x-y} = (2^3)^{x-y} \][/tex]
Using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we can simplify:
[tex]\[ (2^3)^{x-y} = 2^{3 \cdot (x-y)} = 2^{3x - 3y} \][/tex]
Thus:
[tex]\[ 8^{x-y} \can be rewritten as \boxed{2^{3x-3y}} \][/tex]
In summary:
- [tex]\(4^{5x} \can be rewritten as 2^{10x}\)[/tex]
- [tex]\(3^{3x+1} \can be rewritten as 3^{3x} \cdot 3^1\)[/tex]
- [tex]\(8^{x-y} \can be rewritten as 2^{3x-3y}\)[/tex]
### a. Rewrite [tex]\(4^{5x}\)[/tex] using exponent rules
First, recognize that [tex]\(4\)[/tex] can be expressed as [tex]\(2^2\)[/tex]:
[tex]\[ 4 = 2^2 \][/tex]
So, [tex]\(4^{5x}\)[/tex] becomes:
[tex]\[ 4^{5x} = (2^2)^{5x} \][/tex]
Using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we can simplify:
[tex]\[ (2^2)^{5x} = 2^{2 \cdot 5x} = 2^{10x} \][/tex]
Thus:
[tex]\[ 4^{5x} \can be rewritten as \boxed{2^{10x}} \][/tex]
### b. Rewrite [tex]\(3^{3x+1}\)[/tex] using exponent rules
We can use the property of exponents that states [tex]\(a^{m+n} = a^m \cdot a^n\)[/tex]:
[tex]\[ 3^{3x+1} = 3^{3x} \cdot 3^1 \][/tex]
Thus:
[tex]\[ 3^{3x+1}\ can be rewritten as \boxed{3^{3x} \cdot 3^1} \][/tex]
### c. Rewrite [tex]\(8^{x-y}\)[/tex] using exponent rules
First, recognize that [tex]\(8\)[/tex] can be expressed as [tex]\(2^3\)[/tex]:
[tex]\[ 8 = 2^3 \][/tex]
So, [tex]\(8^{x-y}\)[/tex] becomes:
[tex]\[ 8^{x-y} = (2^3)^{x-y} \][/tex]
Using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we can simplify:
[tex]\[ (2^3)^{x-y} = 2^{3 \cdot (x-y)} = 2^{3x - 3y} \][/tex]
Thus:
[tex]\[ 8^{x-y} \can be rewritten as \boxed{2^{3x-3y}} \][/tex]
In summary:
- [tex]\(4^{5x} \can be rewritten as 2^{10x}\)[/tex]
- [tex]\(3^{3x+1} \can be rewritten as 3^{3x} \cdot 3^1\)[/tex]
- [tex]\(8^{x-y} \can be rewritten as 2^{3x-3y}\)[/tex]
