Question

Find the numerical equivalent of 99⋅9−6
. Your answer should be written without exponents

Answer 1

Find the numerical equivalent of 9 to the power of 9 ⋅ 9 to the power of −6 Your answer should be written without exponents

Answer 2

What is the missing number in this simplified exponential expression: 7 to the power of ?÷7 to the power of 5=7 to the power of 12

Answer 3

Apply the Property of Negative Integer Exponents to solve 7 to the power of −3 The answer should not contain any exponent expression or decimals

Answer 4

To find the numerical equivalent of \( 99 \cdot 9^{-6} \), we first need to evaluate \( 9^{-6} \).

The expression \( 9^{-6} \) is the same as \( \frac{1}{9^6} \).

Next, we need to calculate \( 9^6 \):
\[
9^6 = (3^2)^6 = 3^{12}
\]
Calculating \( 3^{12} \):
\[
3^2 = 9, \quad 3^3 = 27, \quad 3^4 = 81, \quad 3^5 = 243, \quad 3^6 = 729
\]
Continuing, we compute \( 3^{12} \) as follows:
\[
3^6 = 729 \quad \Rightarrow \quad 3^{12} = 729 \times 729 = 531441
\]
So, \( 9^6 = 531441 \) and thus:
\[
9^{-6} = \frac{1}{531441}
\]
Now, we multiply:
\[
99 \cdot 9^{-6} = 99 \cdot \frac{1}{531441} = \frac{99}{531441}
\]

Next, we can simplify \( \frac{99}{531441} \):
- Factor \( 99 \):
\[
99 = 3^2 \cdot 11
\]
- Factor \( 531441 \) (as calculated previously, \( 531441 = 9^6 = (3^2)^6 = 3^{12} \)):
\[
531441 = 3^{12}
\]
Now we can rewrite the fraction:
\[
\frac{99}{531441} = \frac{3^2 \cdot 11}{3^{12}} = \frac{11}{3^{10}}
\]
Calculating \( 3^{10} \):
\[
3^{10} = 59049 \quad (\text{since } 3^5 = 243 \text{ and } 3^{10} = 243^2 = 59049)
\]

Thus, the final answer is:
\[
\frac{11}{59049}
\]

So, the numerical equivalent of \( 99 \cdot 9^{-6} \) is:
\[
\boxed{\frac{11}{59049}}
\]

Answer 5

To find the numerical equivalent of \( 9^9 \cdot 9^{-6} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).

Applying this rule, we have:
\[
9^9 \cdot 9^{-6} = 9^{9 + (-6)} = 9^{9 - 6} = 9^3
\]

Now, we calculate \( 9^3 \):
\[
9^3 = 9 \cdot 9 \cdot 9
\]

Calculating step-by-step:
1. First, \( 9 \cdot 9 = 81 \).
2. Then, \( 81 \cdot 9 = 729 \).

Thus, \( 9^3 = 729 \).

Therefore, the numerical equivalent of \( 9^9 \cdot 9^{-6} \) is:
\[
\boxed{729}
\]

Answer 6

To solve the equation

\[
\frac{7^?}{7^5} = 7^{12}
\]

we can use the property of exponents that says \(\frac{a^m}{a^n} = a^{m-n}\). Thus, we rewrite the left-hand side:

\[
7^{? - 5} = 7^{12}
\]

Since the bases are the same (both are base 7), we can set the exponents equal to each other:

\[
? - 5 = 12
\]

Now, we solve for \( ? \):

\[
? = 12 + 5 = 17
\]

Thus, the missing number is

\[
\boxed{17}
\]

Answer 7

To express \( 7^{-3} \) using the property of negative integer exponents, we use the rule that states:

\[
a^{-n} = \frac{1}{a^n}
\]

Applying this to \( 7^{-3} \), we have:

\[
7^{-3} = \frac{1}{7^3}
\]

Now, we need to calculate \( 7^3 \):

\[
7^3 = 7 \cdot 7 \cdot 7
\]

Calculating step-by-step:

1. \( 7 \cdot 7 = 49 \)
2. \( 49 \cdot 7 = 343 \)

Thus,

\[
7^3 = 343
\]

Now substituting back, we find:

\[
7^{-3} = \frac{1}{7^3} = \frac{1}{343}
\]

Since the answer should not contain any exponent expression or decimals, the final result is:

\[
\boxed{\frac{1}{343}}
\]