Apply the properties of integer exponents to generate equivalent expressions to
(14³•27^-⁷)⁰/3⁴ with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To simplify the given expression, let's apply the properties of integer exponents:
1. Product rule: (a^m)(a^n) = a^(m+n)
2. Quotient rule: (a^m)/(a^n) = a^(m-n)
3. Power rule: (a^m)^n = a^(m*n)
First, let's apply the power rule:
(14³ • 27^(-7))^0 / 3^4
Since any non-zero number raised to the power of zero is 1, we have:
1 / 3^4
Now, let's simplify:
1 / 3^4 = 1 / 81
Thus, the simplified expression is 1/81.
To generate equivalent expressions with positive exponents, we can apply the properties of integer exponents. Let's break down the expression step by step:
Step 1: Start with the given expression: (14³•27^-⁷)⁰/3⁴
Step 2: Apply the property that any number raised to the power of 0 equals 1. Therefore, (14³•27^-⁷)⁰ becomes 1.
Step 3: Now we have 1/3⁴. Applying the property that any number with a negative exponent can be reciprocated (flipped) and turned positive, we can rewrite 1/3⁴ as 3^-⁴.
Step 4: We can use the product of powers property to multiply the exponents when we raise one exponent to another exponent. In this case, 3^-⁴ means 3 to the power of -4, which is equivalent to 3^(-1 * 4). Applying the property, we get (3^-1)^4, which simplifies to (1/3)^4.
Step 5: Finally, we simplify (1/3)^4, which means we raise 1/3 to the power of 4. This yields (1/3)*(1/3)*(1/3)*(1/3), or (1/81).
Therefore, the equivalent expression is (14³•27^-⁷)⁰/3⁴ = 1/81.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents. Let's break down the given expression step by step:
Step 1: Simplify the bases of the exponents.
- 14³ = 14 * 14 * 14 = 2,744
- 27^-⁷ = 1 / 27^7 (Since a negative exponent indicates the reciprocal)
Step 2: Combine the two simplified bases using multiplication.
- (14³ • 27^-7)
Step 3: Apply the property of zero exponents.
- Any non-zero number raised to the power of zero is equal to 1.
- (14³ • 27^-7)⁰ = 1
Step 4: Simplify the denominator.
- 3⁴ = 3 * 3 * 3 * 3 = 81
Step 5: Combine the numerator and denominator.
- (14³ • 27^-7)⁰ / 3⁴ = 1 / 81
Step 6: Simplify the fraction.
- 1 / 81 is already simplified, so our final answer is 1/81.
Therefore, the simplified fraction without any remaining exponents is 1/81.