Simplify ((2r^3t)^3)^2

The given problem asks to simplify a mathematical expression involving exponents. The expression provided is a power raised to another power, all of which is applied to a binomial with two variables (r and t, each with their respective exponents) initially multiplied together. The task is to apply the rules of exponents to condense the expression into its simplest algebraic form without actually computing a numerical value.

${(({((2 r^{3} t))}^{3}))}^{2}$

Answer

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Solution:

Step 1:

Apply the exponent multiplication rule to ${({(2 r^{3} t)}^{3})}^{2}$ .

Step 1.1:

Invoke the power of a power rule, which states ${(a^{m})}^{n} = a^{m \cdot n}$ , to the expression ${(2 r^{3} t)}^{3 \cdot 2}$ .

Step 1.2:

Calculate $3 \times 2$ to simplify the expression to ${(2 r^{3} t)}^{6}$ .

Step 2:

Distribute the exponent over the multiplication using the rule ${(a b)}^{n} = a^{n} b^{n}$ .

Step 2.1:

Apply the exponent distribution to $2 r^{3} t$ resulting in ${(2 r^{3})}^{6} t^{6}$ .

Step 2.2:

Further distribute the exponent to $2 r^{3}$ to get $2^{6} {(r^{3})}^{6} t^{6}$ .

Step 3:

Calculate $2^{6}$ which equals $64$ .

Step 4:

Handle the exponent on $r^{3}$ raised to the 6th power.

Step 4.1:

Utilize the power of a power rule again, ${(a^{m})}^{n} = a^{m \cdot n}$ , for $r^{3 \cdot 6}$ .

Step 4.2:

Perform the multiplication $3 \times 6$ to obtain $r^{18}$ , resulting in the final simplified expression $64 r^{18} t^{6}$ .

Knowledge Notes:

The problem involves simplifying an expression with multiple exponents. The key knowledge points used in the solution are:

Power of a Power Rule: This rule states that when you raise a power to another power, you multiply the exponents. For any nonzero number $a$ and integers $m$ and $n$ , the rule is written as $(a^{m})^{n} = a^{m \cdot n}$ .
Product to a Power Rule: This rule allows you to distribute an exponent over a product. For any nonzero numbers $a$ and $b$ and integer $n$ , the rule is written as $(a b)^{n} = a^{n} b^{n}$ .
Exponentiation of Numbers: When you raise a number to a power, you multiply the number by itself as many times as indicated by the exponent. For example, $2^{6}$ means multiplying 2 by itself 6 times, which equals 64.
Combining the Rules: In the given problem, both the power of a power rule and the product to a power rule are used in conjunction to simplify the expression step by step.

Understanding and applying these exponent rules are crucial for simplifying expressions and solving algebraic problems involving exponents.