Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?(1 point)
Responses
only the Power Rule of Exponents
only the Product Rule of Exponents
only the Property of Negative Exponents
the Property of Negative Exponents and the Product Rule of Exponents
The Property of Negative Exponents and the Product Rule of Exponents were used to generate the equivalent expression 3^14 from 3^5/3^−9.
Which of the following is an equivalent expression to 150 ⋅ 7^−2 / (−4)^−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)
To find an equivalent expression with only positive exponents, we can apply the properties of integer exponents.
First, let's simplify the expression step by step:
1. Let's focus on the base 7 and apply the property of negative exponents:
7^(-2) = 1/7^2 = 1/49
2. Next, focus on the base -4 and apply the property of negative exponents:
(-4)^(-3) = 1/(-4)^3 = 1/(-4 × -4 × -4) = 1/(-64) = -1/64
The expression now becomes:
150 ⋅ 1/49 / (-1/64)
Now, let's simplify further:
3. To divide fractions, we can multiply by the reciprocal of the denominator:
150 ⋅ 1/49 * (-64/1)
4. Multiply the numerators and denominators:
(150 * 1 * -64) / (49 * 1)
5. Simplify:
-9600 / 49
Therefore, an equivalent expression with positive exponents is -9600/49.
Which of the following is an equivalent expression to 15^0 ⋅ 7^−2 (−4)^−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)
1/7^2 ⋅ (−4)^−3
(−4)^3/7^2
4^3/7^2
7^2/(−4)^3
To find an equivalent expression with only positive exponents, we can apply the properties of integer exponents.
First, let's simplify the expression step by step:
1. Any number raised to the power of 0 is equal to 1. Therefore, 15^0 = 1.
2. Next, focus on the base 7 and apply the property of negative exponents:
7^(-2) = 1/7^2 = 1/49
3. Next, focus on the base -4 and apply the property of negative exponents:
(-4)^(-3) = 1/(-4)^3 = 1/(-4 × -4 × -4) = 1/(-64) = -1/64
The expression now becomes:
1 ⋅ 1/49 ⋅ -1/64
Now, let's simplify further:
4. Multiply the numerators and denominators:
(1 * 1 * -1) / (49 * 64)
5. Simplify:
-1 / 3136
Therefore, an equivalent expression with positive exponents is -1/3136.
Out of the given options, none of them matches the equivalent expression -1/3136.
Which of the following is the equivalent expression to (15^−3 ⋅ 4^7)^0/4^−3 that has been generated by applying the Properties of Integer Exponents?(1 point)
Responses
60/4^−3
1/64
1⋅4^−3
64
To find the equivalent expression, let's simplify step by step using the properties of integer exponents.
1. First, let's simplify the expression inside the parentheses:
(15^−3 \cdot 4^7) = (1/15^3 \cdot 4^7)
2. Next, simplify the exponents:
1/15^3 = 1/3375
3. Now, let's simplify the expression outside the parentheses:
(1/3375 \cdot 4^7)^0/4^−3
4. Any number (except 0) raised to the power of 0 is equal to 1. Therefore, (1/3375 \cdot 4^7)^0 = 1.
5. Simplify further:
1/4^−3 = 4^3 = 64
So, the equivalent expression is 64.
Out of the given options, the correct answer is 64.
Which of the following is an equivalent expression to (2^−3 ⋅9^3)−4^ / 2^9 ⋅9^−10 with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)
2^3 ⋅9^2
2^3/9^2
2^12⋅9^−12/2^9⋅9^−10
2^−7⋅9^−1/2^9⋅9^−10
To find an equivalent expression with only positive exponents, we can apply the properties of integer exponents.
First, let's simplify the expression step by step:
1. Let's focus on the base 2 and apply the property of negative exponents:
2^(-3) = 1/2^3 = 1/8
2. Next, focus on the base 9 and apply the property of negative exponents:
9^(-10) = 1/9^10
The expression now becomes:
(1/8 * 9^3) - 4^1 / 2^9 * 1/9^10
Now, let's simplify further:
3. Evaluate the exponentials:
(1/8 * 729) - 4 / 512 * 1/9^10
4. Multiply and simplify:
(729/8) - 4 / (512 * 1/9^10)
5. Simplify further:
(729/8) - 4 / (512/9^10)
6. Simplify the exponents of 9:
(729/8) - 4 / (512/1)
7. Simplify further:
(729/8) - 4 / 512
8. Convert 4 to an equivalent with a base of 8:
(729/8) - (2^2 / 512)
9. Simplify further:
(729/8) - (1/64)
10. Find a common denominator for the fractions:
(729/8) - (1/64) = (58368/64) - (1/64)
11. Subtract the fractions:
(58368 - 1) / 64 = 58367 / 64
Out of the given options, none of them matches the equivalent expression 58367/64.