Apply the properties of integer exponents to generate an equivalent to (5^-³)⁶ • 5¹⁷/10^-² with only positive exponents, then solve the expression.
(5^-³)⁶ • 5¹⁷/10^-²=□
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents.
First, let's simplify the expression (5^-³)⁶ • 5¹⁷/10^-²:
(5^-³)⁶ • 5¹⁷/10^-²
The property of raising a power to a power states that (a^m)^n = a^(m * n). Applying this property to (5^-³)⁶:
(5^-³)⁶ = 5^(-³ * 6) = 5^-18
Now the expression becomes:
5^-18 • 5¹⁷/10^-²
The property of dividing powers with the same base states that a^m / a^n = a^(m - n). Applying this property to 5^-18 / 10^-²:
5^-18 / 10^-² = 5^(-18 - (-2)) = 5^-18 + 2 = 5^-16
Now the expression becomes:
5^-16 • 5¹⁷
The property of multiplying powers with the same base states that a^m * a^n = a^(m + n). Applying this property to 5^-16 • 5¹⁷:
5^-16 • 5¹⁷ = 5^(-16 + 17) = 5^1 = 5
Therefore, the simplified expression is 5.
To generate an equivalent expression with only positive exponents, we can use the properties of integer exponents.
First, let's rewrite the given expression:
(5^-³)⁶ • 5¹⁷/10^-²
Using the property (a^m)^n = a^(m*n), we can simplify the exponents:
(5^(-³*6)) • 5^(1*17)/10^(-²)
Simplifying the exponents further:
5^(-³*6) • 5^17/10^(-²)
Using the property a^(-m) = 1/a^m, we can convert the negative exponents to positive exponents:
1/5^(³*6) • 5^17/10^(²)
Now, let's simplify further:
1/5^18 • 5^17/10^2
Using the property a^m/a^n = a^(m-n), we can combine the division:
5^(17-18)/10^2
Simplifying the exponents:
5^-1/10^2
Using the property a^(-m) = 1/a^m again:
1/5^1/10^2
Simplifying the expression:
1/5/100
Using the property of dividing by a fraction (a/b) / c = (a/b) * (1/c), we can divide:
1/5 * 100/1
Simplifying the expression:
100/5
Finding the quotient:
20
Therefore, (5^-³)⁶ • 5¹⁷/10^-² is equivalent to 20.
To generate an equivalent expression with only positive exponents, we can use the property that a negative exponent is the reciprocal of the corresponding positive exponent.
Step 1: Convert negative exponents to positive exponents:
(5^-³)⁶ • 5¹⁷/10^-² is equivalent to (1/5³)⁶ • 5¹⁷/(1/10²)
Step 2: Simplify fractions:
(1/125)⁶ • 5¹⁷/(1/100)
Step 3: Simplify exponents:
(1/125)⁶ • 5¹⁷ • (100/1)
Now, let's solve the expression:
Step 4: Evaluate the exponents:
(1/125)⁶ = 1/125⁶ = 1/244140625
5¹⁷ = 5⁷ • 5¹⁰ = 78125 • 9765625
Step 5: Multiply the numbers:
(1/244140625) • (78125 • 9765625) • (100/1)
Step 6: Simplify the multiplication:
= (78125 • 9765625) / (244140625) • 100
= 762939453125 / 244140625 • 100
= 312500 • 100
= 31250000
Therefore, the expression (5^-³)⁶ • 5¹⁷/10^-² is equal to 31,250,000.