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Which of the following is an equivalent expression to 5^9⋅5^−13 with only positive exponents that has been generated by applying

11 answers

To simplify the expression \( 5^9 \cdot 5^{-13} \) using the laws of exponents, we can apply the rule that states when multiplying expressions with the same base, you add their exponents:

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

In this case, we can apply this rule:

\[ 5^9 \cdot 5^{-13} = 5^{9 + (-13)} = 5^{9 - 13} = 5^{-4} \]

Next, to express \( 5^{-4} \) using only positive exponents, we can use the property that states \( a^{-n} = \frac{1}{a^n} \):

\[ 5^{-4} = \frac{1}{5^4} \]

Thus, the equivalent expression to \( 5^9 \cdot 5^{-13} \) with only positive exponents is:

\[ \frac{1}{5^4} \]

Good, the answer was 1 over 5^4
Next, which property of exponents was used to generate the equivalent expression 3^14 from 3^5 over 3^-9

To generate the equivalent expression \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \), we apply the law of exponents that states when dividing expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator:

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

In this case, we have:

\[ \frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14} \]

So, the property of exponents used to generate the equivalent expression \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \) is the quotient property of exponents.

Okay, which of the following is an expression to 15^0 times 7^-2 over 9-4)^-3 with only positive exponents that has been generated by applying the properties of integer expoents?

To simplify the expression \( \frac{15^0 \cdot 7^{-2}}{(9 - 4)^{-3}} \) and express it with only positive exponents, let's follow these steps:

  1. Simplify \( (9 - 4)^{-3} \): \[ 9 - 4 = 5 \] Therefore, \[ (9 - 4)^{-3} = 5^{-3} \]

  2. Apply the property for exponents: When you have a negative exponent, you can rewrite it as: \[ a^{-n} = \frac{1}{a^n} \] Hence, \[ (9 - 4)^{-3} = 5^{-3} = \frac{1}{5^3} \]

  3. Combine the expressions: The expression now becomes: \[ \frac{15^0 \cdot 7^{-2}}{5^{-3}} \]

  4. Rewrite using positive exponents: We know that \( 15^0 = 1\) (any non-zero number raised to the power of 0 equals 1), so we have: \[ \frac{1 \cdot 7^{-2}}{5^{-3}} = \frac{7^{-2}}{5^{-3}} \]

  5. Use the property of exponents for division: The division of exponents can be rewritten as: \[ \frac{a^m}{b^n} = a^m \cdot b^{-n} \] Therefore: \[ \frac{7^{-2}}{5^{-3}} = 7^{-2} \cdot 5^{3} \]

  6. Convert negative exponents to positive: Applying the rule \( a^{-n} = \frac{1}{a^n} \): \[ 7^{-2} = \frac{1}{7^2} \] Therefore, \[ 7^{-2} \cdot 5^3 = \frac{5^3}{7^2} \]

So the expression \( \frac{15^0 \cdot 7^{-2}}{(9 - 4)^{-3}} \) with only positive exponents is:

\[ \frac{5^3}{7^2} \]

Do you mean...4^3 over 7^2?

I understand the confusion. Let's revisit the expression step-by-step to clarify:

Given the expression:

\[ \frac{15^0 \cdot 7^{-2}}{(9 - 4)^{-3}} \]

  1. Evaluate \( 15^0 \):

    • \( 15^0 = 1 \)

  2. Simplify \( (9 - 4)^{-3} \):

    • \( 9 - 4 = 5 \)
    • So, \( (9 - 4)^{-3} = 5^{-3} \)

Now substituting these into the expression, we have:

\[ \frac{1 \cdot 7^{-2}}{5^{-3}} = \frac{7^{-2}}{5^{-3}} \]

  1. Use the property of exponents for division:

    • Recall that \( \frac{a^m}{b^n} = a^m \cdot b^{-n} \):
    • Thus, this becomes: \[ 7^{-2} \cdot 5^{3} \]

  2. Convert the negative exponent to positive:

    • The expression \( 7^{-2} \) can be rewritten as \( \frac{1}{7^2} \):
    • So, we get: \[ 7^{-2} \cdot 5^3 = \frac{5^3}{7^2} \]

Conclusion:

The fully simplified expression with only positive exponents is

\[ \frac{5^3}{7^2} \]

Now, if you meant to ask about a different expression that specifically involves \( 4^3 \) instead of \( 5^3 \), please provide the original context or expression. Based on the discussion so far, \( 5^3 \) is correct from the calculation above based on the provided expression.

which of the following is the equivalent expression to (15^-3 times 4^7)^0 over 4^-3that has been generated by applying the properties of integer exponents

To simplify the expression

\[ \frac{(15^{-3} \cdot 4^{7})^{0}}{4^{-3}} \]

we can apply the properties of exponents step-by-step.

  1. Apply the Zero Exponent Rule: According to the zero exponent rule, any non-zero base raised to the power of zero is equal to 1. Thus:

    \[ (15^{-3} \cdot 4^{7})^{0} = 1 \]

  2. Substituting into the Expression: Substituting this result back into the original expression gives us:

    \[ \frac{1}{4^{-3}} \]

  3. Rewrite Using Positive Exponents: The term \( 4^{-3} \) can be rewritten as:

    \[ 4^{-3} = \frac{1}{4^{3}} \]

    So,

    \[ \frac{1}{4^{-3}} = 4^{3} \]

Putting all this together, we simplify the original expression to:

\[ 4^{3} \]

Therefore, the equivalent expression to

\[ \frac{(15^{-3} \cdot 4^{7})^{0}}{4^{-3}} \]

using only positive exponents is

\[ 4^{3} \]

which of the following is an equivalent expression to (2^-3 times 9^3)^-4 over 2^9 times 9^-10 with only positive exponents generated by applying the properties of integer exponents

To simplify the expression

\[ \frac{(2^{-3} \cdot 9^{3})^{-4}}{2^{9} \cdot 9^{-10}} \]

and express it using only positive exponents, we'll go through the following steps:

  1. Apply the Power of a Product Rule: According to the properties of exponents, when raising a product to a power, we can apply the exponent to each factor:

    \[ (a \cdot b)^n = a^n \cdot b^n \]

    Thus, we can rewrite the numerator:

    \[ (2^{-3} \cdot 9^{3})^{-4} = (2^{-3})^{-4} \cdot (9^{3})^{-4} \]

    This simplifies further using the power of a power rule:

    \[ = 2^{12} \cdot 9^{-12} \]

  2. Rewrite the Full Expression: Now, substitute the simplified numerator back into the expression:

    \[ \frac{2^{12} \cdot 9^{-12}}{2^{9} \cdot 9^{-10}} \]

  3. Apply the Quotient Rule: When dividing like bases, we subtract the exponents:

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

    First, simplify the \(2\) terms:

    \[ \frac{2^{12}}{2^{9}} = 2^{12 - 9} = 2^{3} \]

    Next, simplify the \(9\) terms:

    \[ \frac{9^{-12}}{9^{-10}} = 9^{-12 - (-10)} = 9^{-2} = 9^{-12 + 10} = 9^{-2} \]

  4. Combine Results: Thus, combining the results:

    \[ 2^{3} \cdot 9^{-2} \]

  5. Convert Negative Exponent: To express \(9^{-2}\) with a positive exponent:

    \[ 9^{-2} = \frac{1}{9^{2}} \]

    Therefore:

    \[ 2^{3} \cdot 9^{-2} = \frac{2^{3}}{9^{2}} \]

Thus, the final equivalent expression in terms of only positive exponents is:

\[ \frac{2^{3}}{9^{2}} \]