Apply the Properties of Integer Exponents to generate equivalent expressions to (143⋅27−7)034 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To apply the properties of integer exponents, we can use the following rules:
1) Product of Powers: (a^m) * (a^n) = a^(m+n)
2) Power of a Power: (a^m)^n = a^(m*n)
3) Power of a Product: (ab)^n = a^n * b^n
4) Negative Exponent: a^(-m) = 1 / a^m
Applying these properties, we can rewrite the expression (143⋅27^(-7))^0.34:
(143⋅27^(-7))^0.34
= (143^(0.34)) * (27^(-7*0.34))
= 143^(0.34) * 27^(-2.38)
Now, we need to simplify this expression. We cannot simplify 143^(0.34) further, but we can simplify 27^(-2.38) by using a negative exponent:
27^(-2.38) = 1 / 27^(2.38)
Thus, the simplified expression is:
143^(0.34) / 27^(2.38)
To further simplify, we can express 143 and 27 as powers of primes:
143 = 11 * 13
27 = 3^3
Now, substitute these prime powers into the expression:
(11 * 13)^(0.34) / (3^3)^(2.38)
= 11^(0.34) * 13^(0.34) / 3^(3*2.38)
Since we cannot simplify further, we can use a calculator to evaluate the expression:
11^(0.34) ≈ 2.905
13^(0.34) ≈ 3.402
3^(3*2.38) ≈ 366.98
Substituting these values, the expression simplifies to:
2.905 * 3.402 / 366.98
Finally, performing the division:
(2.905 * 3.402) / 366.98 ≈ 0.0269
Therefore, the solution to the expression (143⋅27^(-7))^0.34 is approximately 0.0269.
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. Product Rule: (a ⋅ b)^c = a^c ⋅ b^c
2. Power Rule: (a^b)^c = a^(b ⋅ c)
3. Quotient Rule: (a / b)^c = a^c / b^c
Let's apply these properties step by step to the given expression:
(143 ⋅ 27^(-7))^0.34
Step 1: Apply the power rule to 27^(-7):
27^(-7) = (3^3)^(-7) = 3^(3 ⋅ -7) = 3^(-21)
Step 2: Apply the product rule to 143 ⋅ 3^(-21):
143 ⋅ 3^(-21) = 143^1 ⋅ 3^(-21) = (143 ⋅ 3^(-21))^1
Step 3: Apply the power rule to (143 ⋅ 3^(-21))^1:
(143 ⋅ 3^(-21))^1 = 143^1 ⋅ 3^(-21) = 143 ⋅ 3^(-21)
Now, let's simplify the expression:
143 ⋅ 3^(-21) can be written as (143 / 3^21) since the exponent becomes positive in the denominator.
Therefore, the simplified expression is:
(143 ⋅ 27^(-7))^0.34 = 143 / 3^21
We have expressed the original expression using only positive exponents. Now, let's solve it.
143 / 3^21 = 143 / 3^(7 ⋅ 3) = 143 / (3^7)^3 = 143 / 2187^3
Simplifying further, we have:
143 / 2187^3 = 143 / 1353708657
Thus, the solution to the expression is the simplified fraction: 143 / 1353708657.