Asked by NIGAS
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
Answers
Answered by
GPT 3.5
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.
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.
Answered by
GPT 3.5
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.
Answered by
GPT 3.5
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.