Solution: Step 1: Compute the value of 23. We get 23 = 8. Now the expression looks like 8(2m)2. Step 2: Use the power of a product rule, which states (ab)n = an bn, to simplify (2m)2 to 22 m2. Step 3: Calculate the square of 2, which is 22 = 4. Substitute this back into the expression to get 8(4m2). Step 4: Perform the multiplication of the constants 8 and 4 to get 32. The final simplified expression is 32m2. Knowledge Notes: The problem involves simplifying an algebraic expression using exponent rules. Here are the relevant knowledge points: 1. Exponentiation: Raising a number to a power, denoted as an, where a is the base and n is the exponent, means multiplying the base a by itself n times. 2. Power of a Product Rule: This rule states that when two or more factors are raised to a power, the power applies to each factor individually. Mathematically, it is expressed as (ab)n = an bn. 3. Multiplication of Constants: Constants can be multiplied directly. If a and b are constants, then ab is also a constant. In the given problem, the expression 23(2m)2 is simplified by applying these rules. The process involves raising the base 2 to the power of 3 and 2, simplifying the product inside the parentheses, and then multiplying the constants to get the final result.

Simplify 2^3(2m)^2

The problem presented is a mathematical expression that needs to be simplified. It involves the operations of exponentiation and multiplication on numerical and algebraic terms. Specifically, the problem asks for the simplification of the product of 2 raised to the power of 3 and the quantity (2m) raised to the power of 2. The question tests the understanding of the laws of exponents as well as the distributive property when dealing with both constants and variables in algebraic expressions.

$2^{3} {((2 m))}^{2}$

Answer

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Solution:

Step 1:

Compute the value of $2^{3}$ . We get $2^{3} = 8$ . Now the expression looks like $8 (2 m)^{2}$ .

Step 2:

Use the power of a product rule, which states $(a b)^{n} = a^{n} b^{n}$ , to simplify $(2 m)^{2}$ to $2^{2} m^{2}$ .

Step 3:

Calculate the square of $2$ , which is $2^{2} = 4$ . Substitute this back into the expression to get $8 (4 m^{2})$ .

Step 4:

Perform the multiplication of the constants $8$ and $4$ to get $32$ . The final simplified expression is $32 m^{2}$ .

Knowledge Notes:

The problem involves simplifying an algebraic expression using exponent rules. Here are the relevant knowledge points:

Exponentiation: Raising a number to a power, denoted as $a^{n}$ , where $a$ is the base and $n$ is the exponent, means multiplying the base $a$ by itself $n$ times.
Power of a Product Rule: This rule states that when two or more factors are raised to a power, the power applies to each factor individually. Mathematically, it is expressed as $(a b)^{n} = a^{n} b^{n}$ .
Multiplication of Constants: Constants can be multiplied directly. If $a$ and $b$ are constants, then $a b$ is also a constant.

In the given problem, the expression $2^{3} (2 m)^{2}$ is simplified by applying these rules. The process involves raising the base $2$ to the power of $3$ and $2$ , simplifying the product inside the parentheses, and then multiplying the constants to get the final result.