Represent the following expressions as a power of the number a (a≠0):
((a^2)^−2)^5÷(a^4·a)^3
Oh, boy! We're diving into the world of exponents and powers! Hang on tight, and let's unravel this expression together. Here's the given expression:
((a^2)^−2)^5÷(a^4·a)^3
To simplify it and represent it as a power of a, we need to start by simplifying the exponents and combining like terms. So, let's start clowning around with the expression!
First, let's deal with the double negative exponent, because we don't need any negativity right now. When we have ((a^2)^−2), we can translate that to (1/(a^2)^2), which simplifies to 1/a^4. Now we have:
(1/a^4)^5 ÷ (a^4·a)^3
Next, let's focus on the denominator, which is (a^4·a)^3. When we have a multiplication involving exponents, we can simply add them together. In this case, we have a^4·a^3, making it a^7. Now we have:
(1/a^4)^5 ÷ a^7
Since we're dividing, we can subtract the exponents. Dividing (1/a^4) by a^7 is the same as multiplying by a^{-7}, so we get:
(1/a^4)^5 · a^(-7)
Now, multiplying powers with the same base means adding the exponents. So, we have (1^5)/(a^4)^5 · a^(-7), which simplifies to:
1/a^20 · a^(-7)
Hmm, we have a negative exponent. But no worries, we can easily make it positive. Remember, a negative exponent means we take the reciprocal of the number. So, a^(-7) becomes 1/a^7. Now we have:
1/a^20 · 1/a^7
Finally, when we multiply fractions, we multiply the numerators together and the denominators together. So, we get:
1/(a^20·a^7)
Combining like terms, we add the exponents of a, which gives us:
1/a^27
Ta-da! We've arrived at our destination. The given expression ((a^2)^−2)^5÷(a^4·a)^3 can be represented as a power of a, and that power is a^(-27), or if you prefer, 1/a^27. Keep on clowning with those powers, my friend!
To represent the given expressions as a power of the number "a," let's simplify it step by step.
Step 1: Simplify the expression ((a^2)^−2)^5
Inside the parentheses, (a^2)^−2 can be rewritten as 1/(a^2)^2 which is equal to 1/a^4. So, ((a^2)^−2)^5 can be simplified as (1/a^4)^5.
Step 2: Apply the power rule for exponents.
To apply the power rule for exponents, we multiply the exponents. (1/a^4)^5 becomes 1^5 / (a^4)^5, which is simply 1/a^20.
Step 3: Simplify the expression (a^4·a)^3
(a^4·a)^3 can be simplified as a^(4·3)·a^3, which is equal to a^12·a^3.
Step 4: Apply the power rule for exponents.
To apply the power rule for exponents, we add the exponents. Therefore, a^12·a^3 becomes a^(12 + 3), which is equal to a^15.
Step 5: Simplify the expression ((a^2)^−2)^5 ÷ (a^4·a)^3
Now that we have simplified both parts separately, let's simplify the entire expression. ((a^2)^−2)^5 ÷ (a^4·a)^3 can be written as (1/a^20) ÷ a^15.
Step 6: Apply the division rule for exponents.
To apply the division rule for exponents, we subtract the exponents. (1/a^20) ÷ a^15 becomes 1/a^20 ÷ 1/a^15, which is equal to (1/a^20) * (a^15/1).
Step 7: Simplify the expression further.
Multiplying the fractions, (1/a^20) * (a^15/1) can be written as a^15/a^20.
Step 8: Apply the rule for subtracting exponents.
To apply the rule for subtracting exponents, we subtract the exponents in the denominator from the exponents in the numerator. Therefore, a^15/a^20 becomes a^(15 - 20), which is equal to a^(-5).
Therefore, the given expression ((a^2)^−2)^5 ÷ (a^4·a)^3 can be represented as a power of the number "a" as a^(-5).
To represent the given expressions as a power of the number a, we need to simplify them step by step. Let's start:
1. Simplify (a^2)^-2:
To simplify a negative exponent, we can rewrite it as the reciprocal of the positive exponent. Thus, (a^2)^-2 becomes 1/(a^2)^2. This is equivalent to 1/(a^4).
2. Simplify 1/(a^4)^5:
To simplify this expression, we can multiply the exponents inside the parentheses. So, (1/(a^4))^5 is equal to 1/(a^4 * 5). This simplifies further to 1/(a^20).
3. Simplify (a^4 * a)^3:
Here, we have the product of two terms with the same base (a). To multiply the terms, we add the exponents. Therefore, (a^4 * a)^3 simplifies to a^(4 + 1). This becomes a^5.
Now, we can rewrite the entire expression with the simplified forms:
((a^2)^-2)^5 ÷ (a^4 * a)^3 becomes (1/(a^4))^5 ÷ a^5.
Next, we can simplify further by using the exponent properties:
4. Simplify (1/(a^4))^5:
When raising a fraction to a power, we can apply the exponent to both the numerator and denominator. Thus, (1/(a^4))^5 results in 1^5 ÷ (a^4)^5. This simplifies to 1 ÷ a^(4 * 5) or 1 ÷ a^20.
5. Rewrite the entire expression using the simplified form:
(1/(a^4))^5 ÷ a^5 becomes 1 ÷ a^20 ÷ a^5.
To divide exponential terms, we can subtract the exponents. Hence, a^20 ÷ a^5 can be expressed as a^(20 - 5) or a^15.
Therefore, the given expression ((a^2)^-2)^5 ÷ (a^4 * a)^3 can be represented as a^15.
((a^2)^−2)^5÷(a^4·a)^3
= (a^-4)^5÷(a^5)^3
= a^-20 ÷ a^15
you do the last step