Solution: Step 1: Apply the negative exponent rule b-n = frac1bn to the first term. frac16 cdot 4-2 Step 2: Apply the negative exponent rule again to the second term. frac16 cdot frac142 Step 3: Combine the fractions. frac1 cdot 16 cdot 42 Step 4: Proceed to simplify the expression. Step 4.1: Calculate the product of the numerators. frac16 cdot 42 Step 4.2: Compute the value of 42. frac16 cdot 16 Step 4.3: Multiply the denominators. frac196 Step 5: Express the result in various forms. Exact Form: frac196 Decimal Form: 0.0104166667 Knowledge Notes: The problem-solving process involves simplifying an expression with negative exponents. The key knowledge points required to solve this problem include: 1. Negative Exponent Rule: The rule states that b-n = frac1bn, which means that a negative exponent indicates the reciprocal of the base raised to the positive exponent. This rule is used to transform expressions with negative exponents into fractions. 2. Multiplication of Fractions: When multiplying fractions, you multiply the numerators together and the denominators together. The expression fracab cdot fraccd = fraca cdot cb cdot d demonstrates this property. 3. Exponentiation: Raising a number to a power, denoted as bn, means multiplying the base b by itself n times. In this problem, 42 means 4 cdot 4. 4. Simplification: The process of simplifying an expression involves performing arithmetic operations and reducing the expression to its simplest form. In this case, the expression is simplified to a single fraction. 5. Decimal Representation: A fraction can be expressed as a decimal by dividing the numerator by the denominator. This is useful for understanding the value of the fraction in a different form. By understanding these concepts, one can solve a wide range of problems involving exponents and fractions.

Simplify 6^-1(4^-2)

The problem is asking for the simplification of an algebraic expression that involves negative exponents. You are requested to evaluate the expression by applying the rules of exponents, particularly the rule which states that a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. The expression given has two parts, with 6 raised to the power of -1, and 4 raised to the power of -2, and these two are being multiplied together. The aim is to rewrite the expression in a simpler form which typically involves positive exponents and/or fractions, reflecting the multiplication of the reciprocal of the bases with the given negative exponents.

$6^{- 1} (4^{- 2})$

Answer

Expert–verified

Solution:

Step 1:

Apply the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ to the first term.

$\frac{1}{6} \cdot 4^{- 2}$

Step 2:

Apply the negative exponent rule again to the second term.

$\frac{1}{6} \cdot \frac{1}{4^{2}}$

Step 3:

Combine the fractions.

$\frac{1 \cdot 1}{6 \cdot 4^{2}}$

Step 4:

Proceed to simplify the expression.

Step 4.1:

Calculate the product of the numerators.

$\frac{1}{6 \cdot 4^{2}}$

Step 4.2:

Compute the value of $4^{2}$ .

$\frac{1}{6 \cdot 16}$

Step 4.3:

Multiply the denominators.

$\frac{1}{96}$

Step 5:

Express the result in various forms.

Exact Form: $\frac{1}{96}$ Decimal Form: 0.0104166667

Knowledge Notes:

The problem-solving process involves simplifying an expression with negative exponents. The key knowledge points required to solve this problem include:

Negative Exponent Rule: The rule states that $b^{- n} = \frac{1}{b^{n}}$ , which means that a negative exponent indicates the reciprocal of the base raised to the positive exponent. This rule is used to transform expressions with negative exponents into fractions.
Multiplication of Fractions: When multiplying fractions, you multiply the numerators together and the denominators together. The expression $\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$ demonstrates this property.
Exponentiation: Raising a number to a power, denoted as $b^{n}$ , means multiplying the base $b$ by itself $n$ times. In this problem, $4^{2}$ means $4 \cdot 4$ .
Simplification: The process of simplifying an expression involves performing arithmetic operations and reducing the expression to its simplest form. In this case, the expression is simplified to a single fraction.
Decimal Representation: A fraction can be expressed as a decimal by dividing the numerator by the denominator. This is useful for understanding the value of the fraction in a different form.

By understanding these concepts, one can solve a wide range of problems involving exponents and fractions.