Simplify the following.
a) $3 \times a \times 2 \times b$
b) $c^5 \times c$
c) $2 y^4 \times 5 y^3$
d) $3 g h^2 \times 4 g^3 h^3$

Multiply constants and combine variables.
Use the exponent rule x m × x n = x m + n .
Simplify each expression by applying these rules.
The simplified expressions are: a) 6 ab ​ , b) c 6 ​ , c) 10 y 7 ​ , d) 12 g 4 h 5 ​ .

Explanation

Understanding the Problem We are asked to simplify four algebraic expressions. This involves multiplying constants and combining variables using exponent rules.

Simplifying Expression a a) To simplify 3 × a × 2 × b , we multiply the constants 3 and 2, and then combine the variables a and b.

Result of a Multiplying the constants, we have 3 × 2 = 6 . Combining the variables, we have a × b = ab . Therefore, the simplified expression is 6 ab .

Simplifying Expression b b) To simplify c 5 × c , we use the exponent rule x m × x n = x m + n . In this case, c is the same as c 1 , so we have c 5 × c 1 = c 5 + 1 .

Result of b Adding the exponents, we get 5 + 1 = 6 . Therefore, the simplified expression is c 6 .

Simplifying Expression c c) To simplify 2 y 4 × 5 y 3 , we multiply the constants 2 and 5, and then combine the variables y 4 and y 3 using the exponent rule x m × x n = x m + n .

Calculations for c Multiplying the constants, we have 2 × 5 = 10 . Combining the variables, we have y 4 × y 3 = y 4 + 3 .

Result of c Adding the exponents, we get 4 + 3 = 7 . Therefore, the simplified expression is 10 y 7 .

Simplifying Expression d d) To simplify 3 g h 2 × 4 g 3 h 3 , we multiply the constants 3 and 4, and then combine the variables g and h using the exponent rule x m × x n = x m + n .

Calculations for d Multiplying the constants, we have 3 × 4 = 12 . Combining the variables, we have g 1 × g 3 = g 1 + 3 and h 2 × h 3 = h 2 + 3 .

Result of d Adding the exponents, we get 1 + 3 = 4 for g and 2 + 3 = 5 for h . Therefore, the simplified expression is 12 g 4 h 5 .

Final Answer In summary: a) 3 × a × 2 × b = 6 ab b) c 5 × c = c 6 c) 2 y 4 × 5 y 3 = 10 y 7 d) 3 g h 2 × 4 g 3 h 3 = 12 g 4 h 5


Examples
Simplifying algebraic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, if you are calculating the area of a rectangle with sides 3 a and 2 b , the area would be ( 3 a ) × ( 2 b ) = 6 ab . Similarly, in physics, if you are calculating the kinetic energy of an object with mass 2 m and velocity 3 v , the kinetic energy is 2 1 ​ ( 2 m ) ( 3 v ) 2 = 9 m v 2 . These simplifications help in making calculations easier and more efficient.

Answered by GinnyAnswer | 2025-07-03

The simplified expressions are: a) 6 ab , b) c 6 , c) 10 y 7 , d) 12 g 4 h 5 . Each expression is simplified by multiplying constants and using exponent rules for variables. This process is essential for making algebraic calculations easier.
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Answered by Anonymous | 2025-07-04