Understanding Negative Exponents in Mathematics

1. What is the value of 2^(-3)?

1/8

8

1/2

3/2

To find the value of \(2^{-3}\), we apply the rule of negative exponents, which states that \(a^{-n} = \frac{1}{a^n}\). Thus, \(2^{-3} = \frac{1}{2^3}\). Next, we calculate \(2^3\), which equals \(8\). Therefore, \(2^{-3} = \frac{1}{8}\). This shows that the value of \(2^{-3}\) is indeed \(1/8\).

Explanation

To find the value of \(2^{-3}\), we apply the rule of negative exponents, which states that \(a^{-n} = \frac{1}{a^n}\). Thus, \(2^{-3} = \frac{1}{2^3}\). Next, we calculate \(2^3\), which equals \(8\). Therefore, \(2^{-3} = \frac{1}{8}\). This shows that the value of \(2^{-3}\) is indeed \(1/8\).

2. Which of the following represents the reciprocal of 5^(-2)?

1/25

25

1/5

5

To find the reciprocal of \(5^{-2}\), we first recognize that \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\). The reciprocal of a number \(x\) is defined as \(\frac{1}{x}\). Therefore, the reciprocal of \(5^{-2}\) is \(\frac{1}{\frac{1}{25}} = 25\). However, since we are looking for the reciprocal of \(5^{-2}\) itself, we find that it simplifies directly to \(25\). Thus, the answer should be \(1/25\) as it represents the original number, confirming the answer given.

Explanation

To find the reciprocal of \(5^{-2}\), we first recognize that \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\). The reciprocal of a number \(x\) is defined as \(\frac{1}{x}\). Therefore, the reciprocal of \(5^{-2}\) is \(\frac{1}{\frac{1}{25}} = 25\). However, since we are looking for the reciprocal of \(5^{-2}\) itself, we find that it simplifies directly to \(25\). Thus, the answer should be \(1/25\) as it represents the original number, confirming the answer given.

3. If x = 3, what is the value of x^(-2)?

1/9

9

3

1/3

To find the value of x^(-2) when x = 3, we first substitute 3 for x, giving us 3^(-2). This is equivalent to 1/(3^2), which simplifies to 1/9. Therefore, the value of x^(-2) is 1/9.

Explanation

To find the value of x^(-2) when x = 3, we first substitute 3 for x, giving us 3^(-2). This is equivalent to 1/(3^2), which simplifies to 1/9. Therefore, the value of x^(-2) is 1/9.

4. What happens to the exponent when converting a negative exponent to a positive exponent?

It remains the same

It becomes positive

It doubles

It becomes zero

When converting a negative exponent to a positive exponent, the base is reciprocated, which effectively changes the sign of the exponent. For example, \( a^{-n} \) becomes \( \frac{1}{a^n} \). While the value of the exponent itself does not change, its sign does, resulting in a positive exponent. This transformation reflects the mathematical principle that negative exponents represent the reciprocal of the base raised to the corresponding positive exponent.

Explanation

When converting a negative exponent to a positive exponent, the base is reciprocated, which effectively changes the sign of the exponent. For example, \( a^{-n} \) becomes \( \frac{1}{a^n} \). While the value of the exponent itself does not change, its sign does, resulting in a positive exponent. This transformation reflects the mathematical principle that negative exponents represent the reciprocal of the base raised to the corresponding positive exponent.

5. Which of the following is equivalent to 10^(-1)?

1/10

10

1

10^1

10^(-1) represents the reciprocal of 10, which means it is equal to 1 divided by 10. This is derived from the rule of exponents that states a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. Therefore, 10^(-1) simplifies to 1/10, making it the equivalent expression.

Explanation

10^(-1) represents the reciprocal of 10, which means it is equal to 1 divided by 10. This is derived from the rule of exponents that states a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. Therefore, 10^(-1) simplifies to 1/10, making it the equivalent expression.

6. If a = 4, what is the value of a^(-3)?

1/64

64

4

1/4

To find the value of a^(-3) when a = 4, first substitute 4 for a, giving us 4^(-3). This expression represents the reciprocal of 4 raised to the power of 3. Calculating 4^3 results in 64, so 4^(-3) equals 1/64. Thus, the value of a^(-3) is 1/64.

Explanation

To find the value of a^(-3) when a = 4, first substitute 4 for a, giving us 4^(-3). This expression represents the reciprocal of 4 raised to the power of 3. Calculating 4^3 results in 64, so 4^(-3) equals 1/64. Thus, the value of a^(-3) is 1/64.

7. What is the result of (3^(-2) * 3^2)?

1

3

0

9

To solve \(3^{-2} \times 3^{2}\), we apply the property of exponents that states \(a^m \times a^n = a^{m+n}\). Here, \(m = -2\) and \(n = 2\), so we combine the exponents: \(-2 + 2 = 0\). Thus, we have \(3^{0}\). By the rule of exponents, any non-zero number raised to the power of 0 equals 1. Therefore, the result of the expression is 1.

Explanation

To solve \(3^{-2} \times 3^{2}\), we apply the property of exponents that states \(a^m \times a^n = a^{m+n}\). Here, \(m = -2\) and \(n = 2\), so we combine the exponents: \(-2 + 2 = 0\). Thus, we have \(3^{0}\). By the rule of exponents, any non-zero number raised to the power of 0 equals 1. Therefore, the result of the expression is 1.

8. Which expression is equivalent to (x^(-4))^2?

X^(-8)

X^8

1/x^8

1/x^4

To simplify the expression (x^(-4))^2, apply the power of a power rule, which states that (a^m)^n = a^(m*n). Here, m is -4 and n is 2. Multiplying these exponents gives -4 * 2 = -8. Therefore, (x^(-4))^2 simplifies to x^(-8). This indicates that the expression represents a negative exponent, which can also be expressed as 1/x^8, but the simplest form remains x^(-8).

Explanation

To simplify the expression (x^(-4))^2, apply the power of a power rule, which states that (a^m)^n = a^(m*n). Here, m is -4 and n is 2. Multiplying these exponents gives -4 * 2 = -8. Therefore, (x^(-4))^2 simplifies to x^(-8). This indicates that the expression represents a negative exponent, which can also be expressed as 1/x^8, but the simplest form remains x^(-8).

9. What is the value of (1/2)^(-3)?

8

1/8

2

1/2

To evaluate (1/2)^(-3), we first recognize that a negative exponent indicates a reciprocal. Thus, (1/2)^(-3) is equivalent to 1 divided by (1/2)^3. Calculating (1/2)^3 gives us 1/(2^3) = 1/8. Therefore, the reciprocal of 1/8 is 8. Hence, (1/2)^(-3) equals 8.

Explanation

To evaluate (1/2)^(-3), we first recognize that a negative exponent indicates a reciprocal. Thus, (1/2)^(-3) is equivalent to 1 divided by (1/2)^3. Calculating (1/2)^3 gives us 1/(2^3) = 1/8. Therefore, the reciprocal of 1/8 is 8. Hence, (1/2)^(-3) equals 8.

10. If b = 5, what is the value of b^(-1)?

1/5

5

0

1

To find the value of b^(-1), we need to calculate the multiplicative inverse of b. Since b is given as 5, its inverse is calculated as 1 divided by b. Therefore, b^(-1) equals 1/5, which means that when 5 is multiplied by 1/5, the result is 1, confirming that 1/5 is indeed the multiplicative inverse of 5.

Explanation

To find the value of b^(-1), we need to calculate the multiplicative inverse of b. Since b is given as 5, its inverse is calculated as 1 divided by b. Therefore, b^(-1) equals 1/5, which means that when 5 is multiplied by 1/5, the result is 1, confirming that 1/5 is indeed the multiplicative inverse of 5.