Determine if each pair of expressions is equivalent. Explain your reasoning.
a: 1/3^2 and 3^-2
b: 1/4^-3 and 4^-3
My answer: a: 1/3^2 = 1/9 and 3^-2 = 0.111111111; these pair of expressions are not equivalent.
b: 1/4^-3 = 64 and 4^-3 = 0.015625; these pair of expressions are not equivalent.
Need help!
Well, the first pair of expressions is 1/3^2 and 3^-2. Let's see if they're equivalent, shall we?
Now, 1/3^2 is the same as 1/(3 squared), which simplifies to 1/9. On the other hand, 3^-2 means 3 raised to the power of -2. In simple terms, that's basically taking the reciprocal of 3 squared, which is 1/9 as well. So it looks like these expressions are indeed equivalent! Hooray for equality!
Now, let's move on to the second pair of expressions: 1/4^-3 and 4^-3.
1/4^-3 is equivalent to 1/(4 raised to the power of -3). In other words, it's the same as dividing 1 by 4 raised to the power of -3. And when we do the math, we get 64.
On the other hand, 4^-3 means 4 raised to the power of -3, which is basically taking the reciprocal of 4 cubed. Doing some more calculations, we find that it equals 0.015625.
So, it seems that these two expressions are not equivalent. One expression gives us 64, while the other gives us 0.015625. They may belong to different universes, but in math, they're not the same.
I hope that clarifies things for you! If you have any more questions, feel free to ask!
Sure! Let's go through each pair of expressions step by step.
a) To compare the expressions 1/3^2 and 3^-2, we need to simplify them first.
Starting with 1/3^2, we first square 3: 3^2 = 9. Therefore, 1/3^2 becomes 1/9.
Now let's look at 3^-2. The negative exponent means we take the reciprocal of the base raised to the positive exponent. So, 3^-2 is equal to 1/3^2, which we already simplified as 1/9.
Since both expressions simplify to 1/9, the pair of expressions a: 1/3^2 and 3^-2 are equivalent.
b) For the expressions 1/4^-3 and 4^-3, we will follow the same steps as above.
First, let's simplify 1/4^-3. The negative exponent means we take the reciprocal of the base raised to the positive exponent. So, 4^-3 is equal to 1/4^3. Now we need to find 4^3: 4 × 4 × 4 = 64. Therefore, 1/4^-3 simplifies to 1/64.
Now, let's simplify 4^-3. Again, we find the reciprocal of the base raised to the positive exponent. So, 4^-3 is equal to 1/4^3, which we already simplified as 1/64.
Since both expressions simplify to 1/64, the pair of expressions b: 1/4^-3 and 4^-3 are equivalent.
To summarize, the pair of expressions a: 1/3^2 and 3^-2 are equivalent, while the pair of expressions b: 1/4^-3 and 4^-3 are also equivalent.
To determine if two expressions are equivalent, we need to simplify both expressions and compare their values.
a: 1/3^2 and 3^-2
To simplify 1/3^2, we need to evaluate the exponent first. 3^2 equals 9, so 1/3^2 becomes 1/9.
To simplify 3^-2, we need to evaluate the negative exponent. A negative exponent means that we need to take the reciprocal of the base raised to the positive exponent. So, 3^-2 becomes 1/3^2, which is equivalent to 1/9.
Since 1/3^2 is equal to 1/9, and 3^-2 is also equal to 1/9, the pair of expressions (1/3^2 and 3^-2) are indeed equivalent.
b: 1/4^-3 and 4^-3
To simplify 1/4^-3, we need to evaluate the negative exponent. A negative exponent means that we need to take the reciprocal of the base raised to the positive exponent. So, 4^-3 becomes 1/4^3.
To evaluate 4^3, we need to multiply 4 by itself three times: 4 * 4 * 4 = 64. Therefore, 1/4^3 is equal to 1/64.
To simplify 4^-3, we need to evaluate the negative exponent. Similar to above, 4^-3 becomes 1/4^3, which is also equal to 1/64.
Since 1/4^(-3) is equal to 1/64, and 4^-3 is also equal to 1/64, the pair of expressions (1/4^-3 and 4^-3) are indeed equivalent.
In summary:
a: 1/3^2 and 3^-2 are equivalent because both expressions simplify to 1/9.
b: 1/4^-3 and 4^-3 are equivalent because both expressions simplify to 1/64.
Rule for negative exponents:
x^-a = 1/x^a
so: 3^-2 = 1/3^2 , YES
4^-3 = 1/4^3, in the question the result said: 1/4^-3 = 4^-3 , so NO
the last one should have been obvious even without knowing the rule.
btw, 1/9 = .111111.....