To simplify the expression using the properties of integer exponents, let's break it down step by step:
1. Start with 15^0 * 7^-2 / (-4^)^-3:
Since any number (except 0) raised to the power of 0 equals 1, 15^0 simplifies to 1:
1 * 7^-2 / (-4^)^-3.
2. Next, let's consider (-4^)^-3:
(-4^)^-3 can be rewritten as (-4)^(-3) since parentheses to the power of something is equivalent to the number within the parentheses raised to that power:
1 * 7^-2 / (-4)^(-3).
3. Now, apply the rule that states: a^(-b) = 1 / (a^b):
1 * 7^-2 / 1 / (-4)^3.
4. Simplify (-4)^3:
(-4)^3 equals -64, so we have:
1 * 7^-2 / 1 / -64.
5. To divide by a fraction, you can multiply by its reciprocal:
1 * 7^-2 * -64 / 1.
6. To multiply negative exponents, apply the rule: a^-b = 1 / (a^b):
1 * 1 / (7^2) * -64 / 1.
7. Simplify 7^2:
7^2 equals 49, so we're left with:
1 / 49 * -64.
8. Finally, multiply the fractions together, keeping in mind that a negative multiplied by a positive equals a negative:
-64 / 49.
So, an equivalent expression with only positive exponents is -64/49.
which is an equivalent expression to 15^0*7^-2/(-4^)^-3 with only positive exponents that has been generated by applying the propertiesof integer exponents
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thats wrong
Apologies for the mistake. Let's correct it:
To simplify the expression 15^0 * 7^-2 / (-4^)^-3 using the properties of integer exponents, we can follow these steps:
1. Recall that any number (except 0) raised to the power of 0 equals 1. Therefore, 15^0 simplifies to 1.
2. Next, consider the expression (-4^)^-3. Notice that the expression -4^ does not have a specific exponent. Let's assume it has an exponent of 1, which means -4^ is equal to -4. Therefore, (-4^)^-3 can be simplified to (-4)^-3.
3. Apply the rule that states: a^(-b) = 1 / (a^b). Thus, (-4)^-3 becomes 1 / (-4)^3.
4. Simplify (-4)^3, which is -4 * -4 * -4 = -64.
5. Now, substitute the simplified values back into the original expression: 1 * 7^-2 / (-4^)^-3 becomes 1 * 7^-2 / (1 / (-64)).
6. When dividing by a fraction, we can multiply by its reciprocal. Therefore, we have 1 * 7^-2 * (-64) / 1.
7. To convert negative exponents to positive, we apply the rule a^-b = 1 / (a^b). Thus, 7^-2 becomes 1 / 7^2.
8. Simplify 7^2, which is 49. Therefore, we have 1 * (1 / 49) * (-64) / 1.
9. Finally, multiply the fractions together: (1 * -64) / (49 * 1).
10. Simplify the expression -64 / 49 to get the equivalent expression with positive exponents: -64/49.
Therefore, the equivalent expression with only positive exponents, generated by applying the properties of integer exponents, is -64/49.
To simplify the expression 15^0 * 7^-2 / (-4^)^-3 using the properties of integer exponents, we can follow these steps:
1. Recall that any number (except 0) raised to the power of 0 equals 1. Therefore, 15^0 simplifies to 1.
2. Next, consider the expression (-4^)^-3. Notice that the expression -4^ does not have a specific exponent. Let's assume it has an exponent of 1, which means -4^ is equal to -4. Therefore, (-4^)^-3 can be simplified to (-4)^-3.
3. Apply the rule that states: a^(-b) = 1 / (a^b). Thus, (-4)^-3 becomes 1 / (-4)^3.
4. Simplify (-4)^3, which is -4 * -4 * -4 = -64.
5. Now, substitute the simplified values back into the original expression: 1 * 7^-2 / (-4^)^-3 becomes 1 * 7^-2 / (1 / (-64)).
6. When dividing by a fraction, we can multiply by its reciprocal. Therefore, we have 1 * 7^-2 * (-64) / 1.
7. To convert negative exponents to positive, we apply the rule a^-b = 1 / (a^b). Thus, 7^-2 becomes 1 / 7^2.
8. Simplify 7^2, which is 49. Therefore, we have 1 * (1 / 49) * (-64) / 1.
9. Finally, multiply the fractions together: (1 * -64) / (49 * 1).
10. Simplify the expression -64 / 49 to get the equivalent expression with positive exponents: -64/49.
Therefore, the equivalent expression with only positive exponents, generated by applying the properties of integer exponents, is -64/49.
simplify that
Apologies for the confusion. Let's simplify the expression -64/49 further. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator (-64) and the denominator (49) and divide both by the GCD.
The GCD of 64 and 49 is 1. Therefore, we can simplify the fraction as follows:
-64/49 = (-64 ÷ 1) / (49 ÷ 1) = -64/49.
Since there is no common factor greater than 1 between -64 and 49, the fraction -64/49 cannot be simplified any further.
Therefore, the simplified form of the expression is -64/49.
The GCD of 64 and 49 is 1. Therefore, we can simplify the fraction as follows:
-64/49 = (-64 ÷ 1) / (49 ÷ 1) = -64/49.
Since there is no common factor greater than 1 between -64 and 49, the fraction -64/49 cannot be simplified any further.
Therefore, the simplified form of the expression is -64/49.
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