Apply the Properties of Integer Exponents to generate an equivalent expression to (5−3)6⋅ 51710−2
with only positive exponents, then solve the expression. (1 point)
(5−3)6⋅ 51710−2=
Applying the property of integer exponents, we know that (a^b)^c = a^(b*c). Therefore, we can rewrite (5^-3)^6 as 5^(-3*6).
Now, multiplying 517^1 and 10^-2, we use the property a^m * a^n = a^(m + n). This gives us 517^1 * 10^-2 = 517^(1-2) = 517^-1.
Therefore, the equivalent expression is 5^(-3*6) * 517^-1.
To solve this expression, we need to evaluate the exponents.
First, we simplify 5^(-3*6) = 5^-18. Using the property a^(-n) = 1/(a^n), we can rewrite this as 1/(5^18).
Now, we simplify 517^-1 = 1/517.
Therefore, the simplified expression is (1/(5^18)) * (1/517).
To compute this expression, we multiply the numerators and denominators separately:
(1 * 1) / (5^18 * 517)
= 1 / (5^18 * 517)
The final result is 1 / (5^18 * 517).
To apply the properties of integer exponents and generate an equivalent expression with only positive exponents, we can use the following rules:
1. Negative exponents: Any number raised to a negative exponent can be written as the reciprocal of the number raised to the positive exponent.
a^(-n) = 1/(a^n)
2. Product of exponents: When multiplying two powers with the same base, you can add the exponents.
a^m * a^n = a^(m + n)
Applying these rules to the given expression, we have:
(5^(-3))^6 * 5^17 * 10^(-2)
Since we want to get rid of the negative exponents, let's rewrite the expression using the reciprocal property:
(1/5^3)^6 * 5^17 * 10^(-2)
Simplifying the expression inside the parentheses:
(1/125)^6 * 5^17 * 10^(-2)
Now let's apply the product of exponents:
(1/125)^(6) * (5)^(17) * 10^(-2)
Finally, let's evaluate the expression:
(1/125)^(6) = 1/(125^6) = 1/244140625 = 0.000000004096
(5)^(17) = 141,985,600,000,000,000
10^(-2) = 1/10^2 = 1/100 = 0.01
Plugging the values back into the expression:
0.000000004096 * 141,985,600,000,000,000 * 0.01 = 579,189,000
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents. Let's break it down step by step:
Step 1: Apply the power of a product rule. The power of a product rule states that (a * b)^n is equivalent to a^n * b^n.
So, let's apply this rule to the expression (5^-3)^6. To do that, we need to raise 5^-3 to the power of 6:
(5^-3)^6 = 5^(-3 * 6) = 5^-18
Step 2: Apply the power of a power rule. The power of a power rule states that (a^n)^m is equivalent to a^(n * m).
Now, let's apply the power of a power rule to the expression 5^-18. To do that, we need to raise 5 to the power of -18:
5^-18 = (5^1)^(-18) = 5^(-1 * 18) = 5^(-18)
Step 3: Apply the multiplication rule for exponents. The multiplication rule for exponents states that a^n * a^m is equivalent to a^(n + m).
Next, let's multiply the expression 5^(-18) by 5^17:
5^(-18) * 5^17 = 5^(-18 + 17) = 5^-1
Step 4: Simplify the expression.
5^-1 is equivalent to 1/5^1, which is equal to 1/5.
Therefore, the equivalent expression to (5^-3)^6 * 5^17 * 10^-2 with only positive exponents is 1/5.
To solve the expression, you can simply evaluate it:
(5^-3)^6 * 5^17 * 10^-2 = (1/5)^6 * 5^17 * 1/10^2
= 1/5^6 * 5^17 * 1/10^2
= 1/5^(6-17) * 1/10^2
= 1/5^-11 * 1/100
= 5^11 * 1/100
= 48828125/100
= 488281.25