Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3−178−14 with positive exponents only.(1 point)
To apply the property of negative integer exponents to the expression 3^(-17)/8^(-14), we can rewrite it as the reciprocal with positive exponents:
(1/3^17)/(1/8^14)
Now, we can apply the property of negative exponents, which states that a^(-n) = 1/(a^n), where "a" is any non-zero number:
(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)
To apply the properties of integer exponents, we can simplify the expression (7^(-3))^7 * 7^20 / 6^(-2) with only positive exponents.
First, let's simplify (7^(-3))^7 using the property of power of a power, which states that (a^m)^n = a^(m*n):
(7^(-3))^7 = 7^(-3*7) = 7^(-21)
Now, let's simplify the entire expression using the property of dividing powers with the same base, which states that a^m / a^n = a^(m-n):
(7^(-3))^7 * 7^20 / 6^(-2) = 7^(-21) * 7^20 / 6^(-2)
Next, let's simplify the exponents using the property of multiplying powers with the same base, which states that a^m * a^n = a^(m+n):
= 7^(-21 + 20) / 6^(-2)
= 7^(-1) / 6^(-2)
Finally, let's use the property of negative exponents, which states that a^(-n) = 1 / a^n, to rewrite the expression with positive exponents:
= 1 / 7^1 * 6^2
= 1/7 * 6^2
= 1/7 * 36
= 36/7
So, the answer as an improper fraction is 36/7.
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 numbers with the same base, you add the exponents.
To find the numerical equivalent of 12^(-5) * 12^7, we can use the product rule of integer exponents.
First, let's rewrite 12^(-5) as 1/12^5 using the property of negative exponents.
Now we have (1/12^5) * 12^7.
Using the product rule of integer exponents, we add the exponents of 12:
1/12^(5 + 7)
Simplifying the exponent, we have:
1/12^12
To find the numerical equivalent, we can evaluate 12^12:
12^12 = 8,916,100,448
Therefore, the numerical equivalent of 12^(-5) * 12^7 is 1/8,916,100,448.