Problem 104 - Elementary Algebra

Question

Simplify the following problems. $$ \left(3 x y z^{2}\right)\left(2 x^{2} y^{3}\right)\left(4 x^{2} y^{2} z^{4}\right) $$

Step-by-Step Solution

Answer

Based on the step by step solution provided, create a short answer question: Question: Simplify the following expression: $(3 x y z^2)(2 x^2 y^3)(4 x^2 y^2 z^4)$ Answer: $24x^5y^6z^6$

Step 1: Multiply the terms

First, we will multiply the terms within each parenthesis together: $$ (3 x y z^2)(2 x^2 y^3)(4 x^2 y^2 z^4) $$

Step 2: Simplify by combining like terms

Now, we will use the exponent laws to simplify the expression. Recall that when we multiply expressions with the same base, we add the exponents: $$ (3 \times 2 \times 4)(x \times x^2 \times x^2)(y \times y^3 \times y^2)(z^2 \times z^4) $$

Step 3: Perform the operations

Finally, perform the multiplication and addition of exponents to obtain the simplified expression: $$ (24)(x^{1+2+2})(y^{1+3+2})(z^{2+4}) = 24x^5y^6z^6 $$ The simplified expression is: $$ 24x^5y^6z^6 $$

Key Concepts

Exponent Laws

Understanding exponent laws, also known as the laws of indices, is crucial when simplifying algebraic expressions with variables raised to powers. When multiplying like bases, rather than multiplying the bases themselves, we add the exponents. For example, multiplyng \(x^a \times x^b\) results in \(x^{a+b}\), because the base, \(x\), remains the same. Likewise, if you're dividing like bases, you subtract the exponents, turning \(x^a \/ x^b\) into \(x^{a-b}\).

An important point to remember is that any base raised to the power of zero equals one, \(x^0 = 1\), and a base without an exponent implicitly has an exponent of one, \(x = x^1\). When dealing with a product of multiple terms with exponents, as seen in the given exercise, you apply the exponent laws separately for each base, simplifying the expression greatly.

Combining Like Terms

When simplifying algebraic expressions, combining like terms is a method that allows us to simplify expressions to their most basic form. Like terms are terms whose variables (and their exponents) are the same. For instance, \(2x\) and \(5x\) are like terms, and they can be combined to make \(7x\). However, term \(2x\) is not like \(2y\) or \(2x^2\) because the variables or their exponents differ.

In the context of our exercise, after using exponent laws to combine the exponents, we get each variable raised to a single power (after adding exponents), turning the expression into a single product of numerical coefficient and variable terms. This makes the expression neater and more manageable, leading to a more simplified form.

Algebraic Multiplication

Algebraic multiplication involves multiplying variables and coefficients together. When faced with an expression like \(3a \times 4b\), we multiply the numerical coefficients (3 and 4) and the variables (a and b) separately, resulting in \(12ab\). Variables with exponents are approached by keeping in mind the exponent laws.

In the exercise's step by step solution, we first multiplied the numerical coefficients, and then we applied exponent laws to the variables to multiply the like terms. It is essential to work systematically, ensuring that each element of the expression is multiplied correctly. Remembering that multiplication is commutative meaning \(a \times b = b \times a\) can also simplify the process. Since order does not matter, you can rearrange and group like terms together before combining them, which is a useful strategy for lengthy or complex algebraic products.