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.(1 point)
(143⋅27−7)034=
Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(43 ⋅ 7−2)−579 ⋅ 4−13 =
To generate equivalent expressions with only positive exponents, we'll apply the properties of integer exponents to the expression (143⋅27−7)034.
First, let's simplify the exponents using the properties:
(143⋅27−7)034
= (3^2 * 11 * 3^3 - 7)034 [Expanding 143]
= (3^5 * 11 - 7)034 [Simplifying powers of 3]
= (243 * 11 - 7)034
= (2673 - 7)034
= 2666^034
Now, let's evaluate the expression:
2666^034
= (2^3 * 333 - 2)034 [Expanding 2666]
= (2^3 * 333 - 2)034 [Simplifying powers of 2]
= (8 * 333 - 2)034
= (2664 - 2)034
= 2662^034
Therefore, the equivalent expression is 2662^034.
To generate equivalent expressions with positive exponents, we need to apply the properties of integer exponents. The properties state:
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. Quotient of powers: (a^m) / (a^n) = a^(m-n).
5. Power of a quotient: (a/b)^n = a^n / b^n.
Let's apply these properties to (143 * 27^(-7))^0.34:
First, let's deal with the negative exponent. According to property 4, we can rewrite 27^(-7) as 1 / 27^7. Now our expression is:
(143 * (1 / 27^7))^0.34
Now let's apply property 3 to the expression inside the parentheses:
(143 / 27^7)^0.34
Next, let's apply property 2 to the power of 0.34:
(143 / 27^7)^(34/100)
Now, let's simplify the exponent 34/100. We can reduce it by dividing both the numerator and denominator by their greatest common divisor, which is 2. Therefore, we have:
(143 / 27^7)^(17/50)
Finally, we evaluate the expression:
(143 / 27^7)^(17/50) ≈ 0.0003681
So, the simplified fraction without remaining exponents is approximately 0.0003681.
To apply the properties of integer exponents, we can break down the given expression step by step:
1. Expand the expression inside the parentheses: (143⋅27−7)034 = (3819 - 7)034.
2. Simplify the expression within the parentheses: (3819 - 7) = 3812.
3. Raise 3812 to the power of 034: 3812^34.
Since the original expression only contains positive exponents, we do not need to apply any additional properties.
Now, to solve the expression 3812^34, we need to find a way to simplify it without calculating the actual value. One way to do this is by expressing it as a fraction with no remaining exponents.
Let's represent the simplified expression as a fraction:
3812^34 / 1
Now, let's find a way to reduce the exponents. We know that any number raised to the power of 1 is the number itself, so we can rewrite it as:
(3812^17)^2 / 1
Next, we can rewrite this fraction using the property of multiplying exponents:
3812^(17*2) / 1
Now, we can simplify the exponent:
3812^34 / 1
Therefore, the simplified fraction is 3812/1, which can be written as the whole number 3812.