Simplify 6⁴ × 2³ × 6² × 2⁵: Combining Powers with Same Base Numbers

Question

Simplify the following equation:

64×23×62×25= 6^4\times2^3\times6^2\times2^5=

Video Solution

Solution Steps

00:00 Simply
00:03 Let's arrange the exercise so that equal bases are adjacent
00:13 According to laws of exponents, when multiplying powers with the same base (A)
00:17 We get the same base (A) raised to the sum of the exponents (M+N)
00:21 Let's use this formula in our exercise
00:24 Let's compare terms according to the formula and simplify
00:28 Let's keep the base
00:34 Let's sum the exponents
00:53 Let's use the same method to simplify the second base
01:12 And this is the solution to the question

Step-by-Step Solution

To simplify the equation 64×23×62×25 6^4 \times 2^3 \times 6^2 \times 2^5 , we will make use of the rules of exponents, specifically the product of powers rule, which states that when multiplying two powers that have the same base, you can add their exponents.

Step 1: Identify and group the terms with the same base.
In the expression 64×23×62×25 6^4 \times 2^3 \times 6^2 \times 2^5 , group the powers of 6 together and the powers of 2 together:

  • Powers of 6: 64×62 6^4 \times 6^2

  • Powers of 2: 23×25 2^3 \times 2^5

Step 2: Apply the product of powers rule.
According to the product of powers rule, for any real number a a , and integers m m and n n , the expression am×an=am+n a^m \times a^n = a^{m+n} .

Apply this rule to the powers of 6:
64×62=64+2=66 6^4 \times 6^2 = 6^{4+2} = 6^6 .

Apply this rule to the powers of 2:
23×25=23+5=28 2^3 \times 2^5 = 2^{3+5} = 2^8 .

Step 3: Write down the final expression.
Combining our results gives the simplified expression: 66×28 6^6 \times 2^8 .

Therefore, the solution to the problem is 66×28 6^6 \times 2^8 .

Answer

66×28 6^6\times2^8