Use the Power Rule of Exponents to create a simplified equivalent expression for (19^4)^9
To apply the Power Rule of Exponents, we multiply the exponents. Therefore, for (19^4)^9, we multiply 4 and 9:
4 * 9 = 36
This gives us 19^36 as the simplified equivalent expression.
Apply the Property of Negative Integer Exponents to solve 7^-3. The answer should not contain any exponent expression or decimals.
The Property of Negative Integer Exponents states that any non-zero number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent.
In this case, we have 7^-3. Applying the property, we can rewrite this as:
1 / 7^3
Now, simplifying the expression:
1 / (7 * 7 * 7) = 1 / 343
Therefore, the simplified equivalent expression for 7^-3 is 1/343.
Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3^-17/8^-14\
To apply the Property of Negative Integer Exponents, we can rewrite the given expression as:
(8^14 / 3^17)
Using the property, we can swap the bases and change the exponents from negative to positive:
(3^17 / 8^14)
Therefore, an expression equivalent to 3^-17 / 8^-14 is 3^17 / 8^14.
Apply the Property of Negative Integer Exponents to generate an expression equivalent to 3^-17/8^-14 with positive exponents only
To apply the Property of Negative Integer Exponents and generate an expression with positive exponents only, we can rewrite the given expression as:
1 / (3^17 / 8^14)
Now, let's simplify the expression:
1 / (3^17 / 2^28)
To divide by a fraction, we can multiply by its reciprocal:
1 * (2^28 / 3^17)
This gives us the expression:
2^28 / 3^17
Therefore, an expression equivalent to 3^-17 / 8^-14 with positive exponents only is 2^28 / 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.
To apply the Properties of Integer Exponents and generate equivalent expressions with only positive exponents, we'll simplify each part of the expression separately.
First, let's simplify (7^-3)^7. Using the Power of a Power Property, we multiply the exponents:
(7^-3)^7 = 7^(-3 * 7) = 7^-21
Next, let's simplify 7^20. This doesn't need any further simplification.
Lastly, let's simplify 6^-2. Using the Property of Negative Exponents, we rewrite it as the reciprocal with a positive exponent:
6^-2 = 1 / 6^2 = 1 / 36
Now, we can replace the original expression with the simplified equivalents:
(7^-3)^7 ⋅ 7^20 / 6^-2 = 7^-21 * 7^20 / 1/36
Using the Product of Powers Property and simplifying:
7^(-21 + 20) / 1/36 = 7^-1 / 1/36 = 36 / 7
Therefore, the expression (7^-3)^7 ⋅ 7^20 / 6^-2 is equivalent to 36/7.