Apply the Property of Negative Integer Exponents to solve 7^−3 . The answer should not contain any exponent expression or decimals.(1 point)
7^−3 = __
To apply the Property of Negative Integer Exponents, we know that:
a^(-n) = 1/(a^n)
Therefore, 7^(-3) = 1/(7^3)
Simplifying further, 7^(-3) = 1/343
Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3^−17/8^−14 with positive exponents only.(1 point)
3^−17/8^−14= __
To generate an expression equivalent to 3^(-17)/8^(-14) with positive exponents only, we can use the property of negative exponents:
a^(-n) = 1/(a^n)
Applying this property to both terms in the expression, we get:
3^(-17)/8^(-14) = (1/3^17)/(1/8^14)
Now we can simplify this expression further:
(1/3^17)/(1/8^14) = (8^14)/(3^17)
Therefore, an expression equivalent to 3^(-17)/8^(-14) with positive exponents only is:
(8^14)/(3^17)
Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3)^^7 ⋅ 7^20/6^−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point)
(7^−3)^^7 ⋅ 7^20/6^−2= __
To simplify the expression and generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents. Let's break it down step by step:
First, let's focus on (7^(-3))^7. According to the property of integer exponents, when a power is raised to another power, we multiply the exponents. Therefore, (7^(-3))^7 is equal to 7^(-3*7) = 7^(-21).
Next, let's consider 7^20. Since there are no negative exponents in this term, we don't need to do anything.
Lastly, let's look at 6^(-2). To remove the negative exponent, we can use the property of integer exponents which states that a^(-n) = 1/(a^n). So, 6^(-2) is equal to 1/(6^2) = 1/36.
Now, let's substitute these equivalent expressions back into the original expression and simplify further:
(7^(-3))^7 ⋅ 7^20/6^(-2) = 7^(-21) ⋅ 7^20/1/36
We can now multiply the terms with the same base (7) and add the exponents:
7^(-21) ⋅ 7^20 = 7^(-21+20) = 7^(-1)
Since 7^(-1) is equal to 1/7^1 = 1/7, the final result is:
1/7 * (1/36) = 1/(7*36) = 1/252
Therefore, the answer is 1/252.
Apply the product rule of integer exponents to find the numerical equivalent of 12^−5⋅12^7 . Show your work and explain how you used the product rule of integer exponents.(2 points)
The product rule of integer exponents states that when multiplying two expressions with the same base, we add the exponents.
In the given expression 12^(-5)⋅12^7, we have the same base 12. So, according to the product rule of integer exponents, we can add the exponents:
12^(-5)⋅12^7 = 12^(-5+7) = 12^2
To find the numerical equivalent, we evaluate 12 raised to the power of 2:
12^2 = 12 * 12 = 144
Therefore, the numerical equivalent of 12^(-5)⋅12^7 is 144.