Apply the Properties of Integer Exponents to generate equivalent expressions to ((7^(-0))*7^(20))/(6^(-2)) with only positive exponents, then solve the expression. Provide the answer as an improper fraction. (1 point)
((7^(-3))^(7)*7^(20))/(6^(-2))=
Starting with the given expression:
((7^(-0))*7^(20))/(6^(-2))
We can apply the property that any number raised to the power of 0 is equal to 1:
((1)*7^(20))/(6^(-2))
We can also apply the property that when there is a negative exponent in the numerator or denominator, it can be moved to the opposite position and the sign of the exponent changes:
(7^(20))/(6^(2))
Now we can solve the expression:
7^(20) = 7^(10+10) = (7^10)*(7^10)
6^(2) = 6 * 6
Therefore,
((7^(-0))*7^(20))/(6^(-2)) = ((1)*(7^10)*(7^10))/(6 * 6)
= (7^10)*(7^10)/(6 * 6)
In an improper fraction, this would be:
(7^10)*(7^10)/(6 * 6) = (7^10)/(6 * 6)
So the answer in improper fraction form is:
(7^10)/(6 * 6)
To generate equivalent expressions with only positive exponents, we can apply the properties of integer exponents.
1. First, let's simplify the expressions within the numerator and denominator separately.
For the numerator:
a) Combining the exponents inside the parentheses: (7^(-3))^(7) is equivalent to 7^(-3*7) = 7^(-21).
b) Multiplying the exponents outside the parentheses: (7^(-21))*7^(20) = 7^(-21+20) = 7^(-1).
For the denominator, 6^(-2) remains the same.
2. Now, we have the simplified expression: (7^(-1))/(6^(-2)).
To simplify further, we use the property of negative exponents, which states that a^(-n) = 1/(a^n).
3. Applying the property to our expression, we have: (1/7^1)/(1/6^2).
4. Simplifying, we get: (1/7)/(1/36).
Finally, to divide fractions, we multiply the numerator by the reciprocal of the denominator.
5. Multiplying the fractions, we have: (1/7)*(36/1) = 36/7.
Therefore, the answer, in improper fraction form, is 36/7.
To apply the properties of integer exponents to generate equivalent expressions, let's break down the expression step by step:
1. Start with ((7^(-0))*7^(20))/(6^(-2))
2. Recall that any number (except zero) raised to the power of zero is equal to 1. So simplify the first part, (7^(-0)), to 1.
3. Now you have 1 * 7^(20) / (6^(-2))
4. Recall that when you divide two numbers with the same base but different exponents, you subtract the exponents. So simplify the denominator, (6^(-2)), to (1 / 6^(2)).
5. Now you have 1 * 7^(20) / (1 / 6^(2))
6. Simplify the division by multiplying the numerator by the reciprocal of the denominator. The reciprocal of (1 / 6^(2)) is 6^(2).
7. Now you have 1 * 7^(20) * 6^(2)
8. Applying the property that when you raise a power to another power, you multiply the exponents, simplify (7^(20))^7 to 7^(20 * 7).
9. Now you have 1 * 7^(20 * 7) * 6^(2)
10. Multiply the exponents to further simplify: 7^(140) * 6^(2)
11. Now you have the expression (7^(140)) * (6^(2))
To solve this expression, you can calculate the value by substituting the values of 7^(140) and 6^(2) into a calculator or using a software program. The answer will be given as an improper fraction, indicating both the whole number and the fractional part.
Since the expression cannot be simplified further, solving it will require using numerical methods.