Question

Please help me!

1. Simplify 5 to the 5th over 5 to the 8th. (4 points)

1 over 5 to the 3rd power
53
1 over 25
1 over 5 to the negative 3rd

2. Simplify 5 to the negative 4th power over 5 to the 3rd. (4 points)

57
5−1
1 over 5
1 over 5 to the 7th power

3. Simplify 4 to the 7th over 5 squared all raised to the 3rd power. (4 points)

4 to the 10th over 5 to the 5th
4 to the 4th over 5
4 to the 21st over 5 to the 6th
12 to the 7th over 15 squared

4. In which expression should the exponents be multiplied? (4 points)
one fifth to the 2nd times one fifth to the 6th
9 to the 3rd over 9 to the 4th
73 ⋅ 78
(26)−5

5. Which expression is equivalent to (53)−2? (4 points)

1 over 5 to the 3rd
negative 1 over 5 times 5 times 5 times 5 times 5 times 5 times
53
1 over 5 times 5 times 5 times 5 times 5 times 5 times

Answer 1

`\textsf{1.} \quad (1)/(5^5)`

`\textsf{2.} \quad (1)/(5^7)`

`\textsf{3.} \quad (4^(21))/(5^(6))`

`\textsf{4.} \quad \left(2^6\right)^(-5)`

`\textsf{5.} \quad (1)/(5^6)=(1)/(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 )`

Explanation:

Question 1

`\textsf{Given}: \quad (5^5)/(5^8)`

`\textsf{Apply exponent rule} \quad (a^b)/(a^c)=a^(b-c):`

`\implies (5^5)/(5^8)=5^((5-8))=5^(-3)`

`\textsf{Apply exponent rule} \quad a^(-n)=(1)/(a^n):`

`\implies 5^(-3)=(1)/(5^3)`

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Question 2

`\textsf{Given}: \quad (5^(-4))/(5^3)`

`\textsf{Apply exponent rule} \quad (a^b)/(a^c)=a^(b-c):`

`\implies (5^(-4))/(5^3)=5^((-4-3))=5^(-7)`

`\textsf{Apply exponent rule} \quad a^(-n)=(1)/(a^n):`

`\implies 5^(-7)=(1)/(5^7)`

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Question 3

`\textsf{Given}: \quad \left((4^(7))/(5^2)\right)^3`

`\textsf{Apply exponent rule} \quad (a^b)^c=a^(bc):`

`\implies \left((4^(7))/(5^2)\right)^3=(4^((7 \cdot 3)))/(5^((2 \cdot 3)))=(4^(21))/(5^(6))`

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Question 4

`\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`\implies \left((1)/(5)\right)^2 \cdot\left((1)/(5)\right)^6=\left((1)/(5)\right)^(2+6)=\left((1)/(5)\right)^8`

`\textsf{Apply exponent rule} \quad (a^b)/(a^c)=a^(b-c):`

`\implies (9^3)/(9^4)=9^((3-4))=9^(-1)=(1)/(9)`

`\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`\implies 7^3 \cdot 7^8=7^((3+8))=7^(11)`

`\textsf{Apply exponent rule} \quad (a^b)^c=a^(bc):`

`\implies \left(2^6\right)^(-5)=2^((6 \cdot -5))=2^(-30)=(1)/(2^(30))`

Therefore, the expression in which the exponents should be multiplied is (2⁶)⁻⁵.

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Question 5

`\textsf{Given}: \quad \left(5^3\right)^(-2)`

`\textsf{Apply exponent rule} \quad (a^b)^c=a^(bc):`

`\implies \left(5^3\right)^(-2)=5^((3 \cdot -2))=5^(-6)`

`\textsf{Apply exponent rule} \quad a^(-n)=(1)/(a^n):`

`\implies 5^(-6)=(1)/(5^6)=(1)/(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 )`