Why is 15^0 • 7^-2/(-4)^-3 = 1/7^2 • (-4)^3 incorrect?
The expression in question is:
15^0 • 7^-2/(-4)^-3 = 1/7^2 • (-4)^3
Let's simplify each side of the equation separately:
15^0 • 7^-2/(-4)^-3
Since any non-zero number raised to the power of 0 is 1, we can simplify 15^0 to be 1:
1 • 7^-2/(-4)^-3
Next, we need to simplify the negative exponents:
1 • 1/7^2/(-1)^3 • 4^3
When we raise a negative number to an odd power, it retains its negative sign. Therefore, (-1)^3 is equal to -1:
1 • 1/7^2/(-1) • 4^3
Now let's simplify the fractions:
1 • 1/49/(-1) • 4^3
To divide a fraction by -1, we can multiply the numerator by -1:
1 • -1/49 • 4^3
Now let's evaluate the exponent:
1 • -1/49 • 64
Multiplying the numbers, we get:
-1/49 • 64 = -64/49
Therefore, the simplified expression is -64/49, not 1/7^2 • (-4)^3. Thus, the original claim is incorrect.
To explain why the expression is incorrect, let's simplify each side separately.
On the left side of the equation:
15^0 is equal to 1, since any number (except zero) raised to the power of 0 is always 1.
7^-2 is the reciprocal of 7^2, which means 1/7^2.
(-4)^-3 is the reciprocal of (-4)^3, so it becomes 1/(-4)^3.
So, the left side simplifies to (1)(1/7^2)/(1/(-4)^3) = (1/7^2)/1/(-4)^3 = 1/7^2/1/(-4)^3.
Now, let's simplify the right side of the equation:
1/7^2 is already simplified.
(-4)^3 is equal to -64, as -4 raised to the power of 3 equals -4 * -4 * -4 = -64.
So, the right side simplifies to 1/7^2 * (-4)^3 = 1/49 * (-64) = -64/49.
As you can see, the left side of the equation is different from the right side (-64/49 ≠ 1/7^2/1/(-4)^3). Therefore, the expression 15^0 • 7^-2/(-4)^-3 = 1/7^2 • (-4)^3 is incorrect.
To understand why the expression 15^0 • 7^-2/(-4)^-3 is not equal to 1/7^2 • (-4)^3, we need to follow the rules of exponentiation and order of operations.
Let's break down the expression step by step:
1. Start with 15^0 • 7^-2/(-4)^-3.
2. First, we evaluate the exponents:
- Any number raised to the power of 0 is equal to 1. Therefore, 15^0 becomes 1.
- Similarly, 7^-2 becomes 1/7^2 because any number raised to a negative power can be reciprocated.
After evaluating the exponents, we have: 1 • 1/7^2/(-4)^-3.
3. Next, we evaluate the negative exponents:
- (-4)^-3 means the reciprocal of (-4)^3, so we can rewrite it as 1/(-4)^3.
Now, our expression becomes: 1 • 1/7^2 / 1/(-4)^3.
4. We simplify the division:
- Dividing by a fraction is equivalent to multiplying by its reciprocal. So, we can rewrite 1/7^2 / 1/(-4)^3 as 1/7^2 * (-4)^3.
The expression now becomes: 1 • 1/7^2 * (-4)^3.
5. Finally, we evaluate the remaining exponents:
- 1/7^2 means the reciprocal of 7^2, so it can be written as 1/(7^2).
- (-4)^3 is equal to -4 * -4 * -4, which evaluates to -64.
The expression becomes: 1 • 1/(7^2) * (-64).
When we multiply 1 • 1/(7^2) * (-64), we get the correct calculated value.
Therefore, the expression 15^0 • 7^-2/(-4)^-3 is not equal to 1/7^2 • (-4)^3 because the order of operations and the simplification of each term were not followed correctly in the second expression.