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

Answer 1

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 2

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

Answer 3

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 4

the answer is 729?

Answer 5

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 6

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

Answer 7

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 8

Simplify 1287÷1225
.(1 point)

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 \( \frac{1287}{1225} \).