Question

Use the Product Rule of Exponents to simplify 510⋅55 (1 point)Responses255025 superscript 50 baseline251525 superscript 15 baseline5155 superscript 15 baseline550

Answer 1

Find the numerical equivalent of 99⋅9−6
. Your answer should be written without exponents. (1 point)

Answer 2

the answer is 729?

Answer 3

What is the missing exponent in the following equation?
h450h? = h215
(1 point)

Answer 4

Simplify 1287÷1225
.(1 point)

Answer 5

To simplify \( 5^{10} \cdot 5^{5} \) using the Product Rule of Exponents, which states that \( a^{m} \cdot a^{n} = a^{m+n} \), we proceed as follows:

\[
5^{10} \cdot 5^{5} = 5^{10 + 5} = 5^{15}
\]

So the simplified expression is \( 5^{15} \).

Therefore, the answer is:

**\( 5^{15} \)**

Answer 6

To find the numerical equivalent of \( 9^9 \cdot 9^{-6} \), we can use the Product Rule of Exponents:

\[
a^m \cdot a^n = a^{m+n}
\]

In this case:

\[
9^9 \cdot 9^{-6} = 9^{9 + (-6)} = 9^{3}
\]

Now we simplify \( 9^{3} \):

\[
9^{3} = 9 \cdot 9 \cdot 9 = 729
\]

So, the numerical equivalent of \( 9^9 \cdot 9^{-6} \) is **729**.

Answer 7

Yes, the answer is indeed **729**.

Here’s the step-by-step breakdown again for clarity:

1. Start with the expression: \( 9^9 \cdot 9^{-6} \).
2. Apply the Product Rule of Exponents:
\[
9^9 \cdot 9^{-6} = 9^{9 + (-6)} = 9^{3}.
\]
3. Calculate \( 9^{3} \):
\[
9^{3} = 9 \cdot 9 \cdot 9 = 81 \cdot 9 = 729.
\]

So the final numerical equivalent is indeed **729**.

Answer 8

To find the missing exponent in the equation \( h^{450} \cdot h^{?} = h^{215} \), we can use the Product Rule of Exponents:

\[
a^m \cdot a^n = a^{m+n}
\]

Applying this rule here, we have:

\[
h^{450} \cdot h^{?} = h^{450 + ?}
\]

Setting this equal to \( h^{215} \):

\[
h^{450 + ?} = h^{215}
\]

Since the bases \( h \) are the same, we can equate the exponents:

\[
450 + ? = 215
\]

To find the missing exponent (?), we can solve for it:

\[
? = 215 - 450
\]

\[
? = -235
\]

So, the missing exponent is **\(-235\)**.

Answer 9

To simplify \( \frac{1287}{1225} \), we can start by finding the greatest common divisor (GCD) of the two numbers.

1. **Prime factorization of 1287:**
- To find the prime factors, we can perform successive divisions:
- \( 1287 \) is odd, so it is not divisible by \( 2 \).
- The sum of the digits of \( 1287 \) is \( 1 + 2 + 8 + 7 = 18 \), which is divisible by \( 3 \):
\[
1287 \div 3 = 429
\]
- Now, find the prime factors of \( 429 \):
- \( 429 \div 3 = 143 \) (since \( 4 + 2 + 9 = 15 \), which is divisible by \( 3 \)).
- Now, factor \( 143 \):
- \( 143 \div 11 = 13 \). Thus, \( 143 = 11 \times 13 \).
- So, \( 1287 = 3^2 \times 11 \times 13 \).

2. **Prime factorization of 1225:**
- \( 1225 \) is odd, so it is not divisible by \( 2 \).
- The sum of the digits \( 1 + 2 + 2 + 5 = 10 \) is not divisible by \( 3 \).
- \( 1225 \) ends in \( 5 \), so it is divisible by \( 5 \):
\[
1225 \div 5 = 245
\]
- Next, factoring \( 245 \):
\[
245 \div 5 = 49
\]
\[
49 = 7^2
\]
- So, \( 1225 = 5^2 \times 7^2 \).

3. **Putting it together:**
- We have:
\[
1287 = 3^2 \times 11 \times 13
\]
\[
1225 = 5^2 \times 7^2
\]

4. **Finding common factors:**
- The GCD of \( 1287 \) and \( 1225 \) is \( 1 \) (no common factors).

5. **So, the simplified form of \( \frac{1287}{1225} \) is just \( \frac{1287}{1225} \) since they share no common factors:**

The simplified fraction is:

\[
\frac{1287}{1225}
\]

To provide the answer in its original or simplest form:

**The result is \( \frac{1287}{1225} \), which cannot be simplified further as both numbers are coprime.**

If you need it in decimal form, you can calculate:

\[
1287 \div 1225 \approx 1.05
\]

But as a fraction, it remains \( \frac{1287}{1225} \).