Section 2.3 - Power Of A Power

1) Express (4^4)^4 as a power with a single exponent

4^256

4^16

4^8

4^0

To express (4^4)^4 as a power with a single exponent, we need to multiply the exponents. In this case, the exponent 4 is being raised to the power of 4. So, 4^4 is 256. Therefore, (4^4)^4 is equal to 256^4, which simplifies to 4^16.

Explanation

To express (4^4)^4 as a power with a single exponent, we need to multiply the exponents. In this case, the exponent 4 is being raised to the power of 4. So, 4^4 is 256. Therefore, (4^4)^4 is equal to 256^4, which simplifies to 4^16.

2) Which of the following expression represents the Exponent Principle for Power of a Power?

(a^m)^n = a^(m+n)

(a^m)^n = a^(m-n)

(a^m)^n = a^(m^n)

(a^m)^n = a^(mn)

The correct answer is (a^m)^n = a^(mn). This expression represents the Exponent Principle for Power of a Power. According to this principle, when a power is raised to another power, we multiply the exponents. In this case, we have (a^m)^n, which means we have a base of 'a' raised to the power of 'm', and that entire expression is raised to the power of 'n'. So, we multiply the exponents and get a^(mn) as the final result.

Explanation

The correct answer is (a^m)^n = a^(mn). This expression represents the Exponent Principle for Power of a Power. According to this principle, when a power is raised to another power, we multiply the exponents. In this case, we have (a^m)^n, which means we have a base of 'a' raised to the power of 'm', and that entire expression is raised to the power of 'n'. So, we multiply the exponents and get a^(mn) as the final result.

3) Simplify (4^1)^2(4^3)

4^5 or 1024

4^6 or 4096

4^7 or 16 384

4^9 or 262 144

The expression (4^1)^2(4^3) can be simplified by applying the exponent rules. First, we simplify the exponent (4^1)^2, which is equal to 4^2 or 16. Then, we multiply this result by 4^3, which is equal to 64. Therefore, the final result is 16 * 64 = 1024, which matches the first option.

Explanation

The expression (4^1)^2(4^3) can be simplified by applying the exponent rules. First, we simplify the exponent (4^1)^2, which is equal to 4^2 or 16. Then, we multiply this result by 4^3, which is equal to 64. Therefore, the final result is 16 * 64 = 1024, which matches the first option.

4) Simplify [(3^3)^2] / (3^2)

3^2 or 9

3^3 or 27

3^4 or 81

3^5 or 243

The given expression can be simplified by using the exponent rule that states that when raising a power to another power, you multiply the exponents. Thus, (3^3)^2 can be simplified to 3^6. Dividing this by 3^2 gives us 3^4, which is equal to 81. Therefore, the correct answer is 81.

Explanation

The given expression can be simplified by using the exponent rule that states that when raising a power to another power, you multiply the exponents. Thus, (3^3)^2 can be simplified to 3^6. Dividing this by 3^2 gives us 3^4, which is equal to 81. Therefore, the correct answer is 81.

5) Simplify [(r^6)^4] / [(r^8)]

R^2

R^3

R^12

R^16

To simplify the given expression, we apply the exponent rule which states that when raising a power to another power, we multiply the exponents. In this case, we have (r^6)^4, which simplifies to r^24. Then, we divide r^24 by r^8, which gives us r^(24-8) = r^16. Therefore, the correct answer is r^16.

Explanation

To simplify the given expression, we apply the exponent rule which states that when raising a power to another power, we multiply the exponents. In this case, we have (r^6)^4, which simplifies to r^24. Then, we divide r^24 by r^8, which gives us r^(24-8) = r^16. Therefore, the correct answer is r^16.

6) Simplify [(3^4)(3^3)]^2 / [(3^2)(3^3)]^2

3^2

3^4

3^12

3^16

The given expression can be simplified using the rules of exponents. When we multiply two numbers with the same base, we add their exponents. Therefore, [(3^4)(3^3)]^2 can be simplified to 3^(4+3)^2. Similarly, [(3^2)(3^3)]^2 can be simplified to 3^(2+3)^2. Simplifying further, we get 3^7^2 / 3^5^2. According to the rule of division, when we divide two numbers with the same base, we subtract their exponents. Therefore, 3^7^2 / 3^5^2 simplifies to 3^(7-5)^2 which equals 3^2.

Explanation

The given expression can be simplified using the rules of exponents. When we multiply two numbers with the same base, we add their exponents. Therefore, [(3^4)(3^3)]^2 can be simplified to 3^(4+3)^2. Similarly, [(3^2)(3^3)]^2 can be simplified to 3^(2+3)^2. Simplifying further, we get 3^7^2 / 3^5^2. According to the rule of division, when we divide two numbers with the same base, we subtract their exponents. Therefore, 3^7^2 / 3^5^2 simplifies to 3^(7-5)^2 which equals 3^2.

7) Express (5^5)^3 as a power with a single exponent

5^2

5^8

5^15

5^125

To express (5^5)^3 as a power with a single exponent, we need to multiply the exponents. So, (5^5)^3 can be rewritten as 5^(5*3), which simplifies to 5^15.

Explanation

To express (5^5)^3 as a power with a single exponent, we need to multiply the exponents. So, (5^5)^3 can be rewritten as 5^(5*3), which simplifies to 5^15.

8) Simplify [(z^4)^3] [(z^8)^2]

Z^17

Z^28

Z^70

Z^192

The given expression can be simplified by applying the power of a power rule. First, we simplify each term inside the parentheses by multiplying the exponents. (z^4)^3 becomes z^12, and (z^8)^2 becomes z^16. Then, we multiply the two simplified terms together by adding the exponents. z^12 * z^16 equals z^28. Therefore, the correct answer is z^28.

Explanation

The given expression can be simplified by applying the power of a power rule. First, we simplify each term inside the parentheses by multiplying the exponents. (z^4)^3 becomes z^12, and (z^8)^2 becomes z^16. Then, we multiply the two simplified terms together by adding the exponents. z^12 * z^16 equals z^28. Therefore, the correct answer is z^28.

9) Simplify [(4w^2)]^5 / [(4^2)w]^2

4w^8

4w^4

16w^8

16w^4

To simplify the expression, we can use the rule of exponents. First, we simplify the numerator by raising 4w^2 to the power of 5, which gives us (4^5)(w^10). Next, we simplify the denominator by raising (4^2)w to the power of 2, which gives us (4^4)(w^2). Finally, we divide the numerator by the denominator, which gives us (4^5)(w^10) / (4^4)(w^2). Simplifying further, we can cancel out the common factors of 4 and w^2, leaving us with (4^1)(w^8). Therefore, the simplified expression is 4w^8.

Explanation

To simplify the expression, we can use the rule of exponents. First, we simplify the numerator by raising 4w^2 to the power of 5, which gives us (4^5)(w^10). Next, we simplify the denominator by raising (4^2)w to the power of 2, which gives us (4^4)(w^2). Finally, we divide the numerator by the denominator, which gives us (4^5)(w^10) / (4^4)(w^2). Simplifying further, we can cancel out the common factors of 4 and w^2, leaving us with (4^1)(w^8). Therefore, the simplified expression is 4w^8.