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.

Answer 1

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.