Write each of the following expressions as a power of 2: 4^−6⋅ 4^4⋅ (2^3⋅ 2^−4)^−1
Apply exponent rule:
( a^b )^c = a^bc
4 = 2^2
4^( - 6 ) = ( 2^2 )^( - 6 ) = 2^( - 12 )
4^4 = ( 2^2 )^4 = 2^8
2^3 ∙ 2^( - 4 ) = 2^( - 1 )
( 2^3 ∙ 2^( - 4 ) ) ^( - 1) = ( 2^( - 1 ) )^( - 1 ) = 2^1 = 2
4^( - 6 ) ∙ 4^4 ∙ 2^3 ∙ 2^( - 4 ) ) ^( - 1) = 2^( - 12 ) ∙ 2^8 ∙ 2^1 = 2^( - 3 )
Thank You
Well, well, well, let's see what we have here. We need to simplify each of these expressions and write them as powers of 2. Get ready for a whirlwind of math fun!
Let's start with the first expression: 4^(-6). Now, remember that any number raised to the power of -n is just 1 over that number raised to the power of n. So, 4^(-6) is simply 1 over 4^6. But 4^6 = 4 × 4 × 4 × 4 × 4 × 4 = 2^6 × 2^6 = 2^(6+6) = 2^12. Therefore, 4^(-6) = 1/2^12.
Moving on to the second expression: 4^4. This is just 4 × 4 × 4 × 4 = 2^2 × 2^2 × 2^2 × 2^2 = 2^(2+2+2+2) = 2^8.
Now, let's dive into the final expression: (2^3 × 2^(-4))^(-1). Inside the parentheses, we can simplify to 2^(3-4) = 2^(-1). Then, raising this to the power of -1 gives us 1/2^(-1) = 1/(1/2) = 2.
So, putting it all together, our final expression is 1/2^12 × 2^8 × 2 = 2^(-12) × 2^8 × 2 = 2^(-4) × 2 = 2^(2-4) = 2^(-2).
There you have it! The final expression 4^(-6) × 4^4 × (2^3 × 2^(-4))^(-1) can be written as 2^(-2).
We can use the properties of exponents to simplify each expression.
First, let's simplify 4^−6:
Recall that any number raised to the power of -n is equal to 1 divided by that number raised to the power of n. Therefore,
4^−6 = 1/4^6 = 1/4096.
Next, let's simplify 4^4:
4^4 = 2^8 = 256.
Now, let's simplify (2^3⋅ 2^−4)^−1:
Using the properties of exponents, when we have a power raised to another power, we multiply the exponents. Therefore,
(2^3⋅ 2^−4)^−1 = 2^(3 * -1 + -4) = 2^(-3 - 4) = 2^-7.
Finally, let's combine the simplified expressions:
4^−6⋅ 4^4⋅ (2^3⋅ 2^−4)^−1 = (1/4096) * 256 * 2^-7.
Since 256 is a power of 2 and 2^-7 is already in the form of a power of 2, we can rewrite the expression as:
(1/4096) * (2^8) * (2^-7) = (2^(8 - 7)) / 4096 = 2/4096.
Therefore, the expression 4^−6⋅ 4^4⋅ (2^3⋅ 2^−4)^−1 can be written as the power of 2 as 2/4096.
To write each of the expressions as a power of 2, we need to simplify them using the properties of exponents.
Let's break down each expression one by one:
1. 4^(-6):
The negative exponent indicates that we need to calculate the reciprocal of the base raised to the positive exponent. So, 4^(-6) = 1/(4^6) = 1/4096.
2. 4^4:
This can be rewritten as a power of 2 by using the fact that 4 is equal to 2^2. So, 4^4 = (2^2)^4 = 2^(2*4) = 2^8 = 256.
3. (2^3 * 2^(-4))^(-1):
Inside the brackets, we have a product of two powers of 2. Using the property of exponents, when you multiply two powers with the same base, you add their exponents. So, 2^3 * 2^(-4) = 2^(3 + (-4)) = 2^(-1).
Now, we have (2^(-1))^(-1). Applying the property of exponentiation where raising a power to a negative exponent equals the reciprocal of that power, we can rewrite it as 1/(2^(-1)). Using the property of exponents again, we know that a negative exponent indicates reciprocal, so 2^(-1) = 1/2.
Therefore, (2^(-1))^(-1) = 1/(1/2) = 2.
Now we can put everything together:
4^(-6) * 4^4 * (2^3 * 2^(-4))^(-1) = (1/4096) * 256 * 2
We can simplify further:
(1/4096) * 256 * 2 = (256 * 2) / 4096 = 512 / 4096 = 1/8
Hence, the expression can be written as a power of 2 as 1/8.