Amy, Lin, and Miguel were practicing creating equivalent expression for (15^3)^9. Amy got an expression of 15^27, Miguel got an expression of 15^12, and Lin got an expression of 15^3. Who correctly used the Power Rule of Exponents to create an equivalent expression?

Question

Amy
Miguel
Lin
Amy and Miguel

Answer 1

To determine who correctly used the Power Rule of Exponents to create an equivalent expression for \((15^3)^9\), we need to apply the Power Rule, which states that \((a^m)^n = a^{m \cdot n}\).

For the expression \((15^3)^9\):

1. According to the Power Rule: \[ (15^3)^9 = 15^{3 \cdot 9} = 15^{27}. \]

Now let's evaluate the results of each person:

• Amy got \(15^{27}\) (Correct).
• Miguel got \(15^{12}\) (Incorrect).
• Lin got \(15^{3}\) (Incorrect).

Since only Amy used the Power Rule correctly, the answer is:

Amy.

Answer 2

When applying the Property of Negative Integer Exponents to solve 9^-2, which of the following is the correct answer?
1/2^9
81
1/81
1/9^-2

Answer 3

To solve \(9^{-2}\) using the Property of Negative Integer Exponents, we can apply the rule that states \(a^{-n} = \frac{1}{a^{n}}\).

So for \(9^{-2}\):

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

Now, we can evaluate \(9^{2}\):

\[ 9^{2} = 81. \]

Thus, substituting back gives:

\[ 9^{-2} = \frac{1}{81}. \]

From the choices given:

1. \( \frac{1}{2^9} \) (incorrect)
2. \( 81 \) (incorrect)
3. \( \frac{1}{81} \) (correct)
4. \( \frac{1}{9^{-2}} \) (incorrect)

The correct answer is:

\(\frac{1}{81}\).

Answer 4

Which of the following is an equivalent expression to 11^-13 times 11^-7 with only positive exponents, generated by applying the properties of exponents?
1/11^6
1/11^20
1/22^6
1/22^20

Answer 5

To simplify the expression \(11^{-13} \times 11^{-7}\) using the properties of exponents, we use the property that states \(a^m \times a^n = a^{m+n}\).

So we can combine the exponents:

\[ 11^{-13} \times 11^{-7} = 11^{-13 + (-7)} = 11^{-20}. \]

Now, to express this with only positive exponents, we apply the rule \(a^{-n} = \frac{1}{a^{n}}\):

\[ 11^{-20} = \frac{1}{11^{20}}. \]

Now, let's evaluate the choices provided:

1. \( \frac{1}{11^6} \) (incorrect)
2. \( \frac{1}{11^{20}} \) (correct)
3. \( \frac{1}{22^6} \) (incorrect)
4. \( \frac{1}{22^{20}} \) (incorrect)

The correct equivalent expression is:

\(\frac{1}{11^{20}}\).

Answer 6

Which of the following is the equivalent expression to (13^-7)^0 times 7^-11/7^-9, generated by applying the properties of exponents?
7^11/7^9
13 times 7^9/7^11
13^-7/7^2
1/49

Answer 7

Let's simplify the expression \((13^{-7})^0 \times \frac{7^{-11}}{7^{-9}}\) using the properties of exponents.

1. Simplifying \((13^{-7})^0\): According to the zero exponent rule, any non-zero number raised to the power of 0 is equal to 1: \[ (13^{-7})^0 = 1. \]

2. Simplifying \(\frac{7^{-11}}{7^{-9}}\): We can use the quotient property of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\): \[ \frac{7^{-11}}{7^{-9}} = 7^{-11 - (-9)} = 7^{-11 + 9} = 7^{-2}. \]

3. Combining everything: Now, substitute back into the expression: \[ 1 \times 7^{-2} = 7^{-2}. \]

4. Expressing \(7^{-2}\) with positive exponents: We use the property \(a^{-n} = \frac{1}{a^n}\): \[ 7^{-2} = \frac{1}{7^2}. \]

Since \(7^2 = 49\), we have: \[ 7^{-2} = \frac{1}{49}. \]

Now, let's evaluate the choices provided:

1. \( \frac{7^{11}}{7^{9}} \) (incorrect)
2. \( 13 \times \frac{7^{9}}{7^{11}} \) (incorrect)
3. \( \frac{13^{-7}}{7^2} \) (incorrect)
4. \( \frac{1}{49} \) (correct)

The correct equivalent expression is:

\(\frac{1}{49}\).

Answer 8

Simply u^75/y^30.
45
uy^45
u^45
u^75/y^30

Answer 9

To simplify the expression \(\frac{u^{75}}{y^{30}}\), we'll analyze the expression as it stands, since it is already in a fractional form.

1. The expression \(\frac{u^{75}}{y^{30}}\) contains \(u^{75}\) in the numerator and \(y^{30}\) in the denominator.
2. There are no like terms to combine or further simplifications that can be done without additional operations.

So, \(\frac{u^{75}}{y^{30}}\) does not simplify further into other forms like \(45\), \(uy^{45}\), or \(u^{45}\).

Thus the simplified expression remains:

\[ \frac{u^{75}}{y^{30}}. \]

Among the options provided, the correct answer is:

\(\frac{u^{75}}{y^{30}}\).