Problem 37 - Elementary Algebra

Question

For the following problems, expand the quantities so that no exponents appear. $$ 8 x^{3} y^{2} $$

Step-by-Step Solution

Answer

Answer: The expanded form of the expression $$8x^3y^2$$ is $$8 * x * x * x * y * y$$.

Step 1: STEP 1: Expand x

To expand x, we need to raise it to the power of 3. That means we have to multiply x by itself 3 times: $$x^3 = x * x * x$$.

Step 2: STEP 2: Expand y

Similarly, to expand y, we need to raise it to the power of 2. That means we have to multiply y by itself 2 times: $$y^2 = y * y$$.

Step 3: STEP 3: Combine expanded powers of x and y

Now that we have expanded both x and y, we need to combine them with the coefficient 8. The given expression is $$8x^3y^2$$, so we multiply the coefficient 8 with the expanded powers of x and y: $$8 * (x * x * x) * (y * y)$$.

Step 4: STEP 4: Simplify the expression

Finally, we can simplify the expression by writing it without using any parentheses: $$8 * x * x * x * y * y$$. Therefore, the expanded expression with no exponents is $$8x^3y^2 = 8 * x * x * x * y * y$$.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising one number, the base, to the power of another number, the exponent. In algebra, exponentiation is used to represent repeated multiplication of a base number. For instance, when we have an expression like \( x^3 \), it is shorthand for multiplying \( x \) by itself three times: \( x \times x \times x \). The small number '3' here is called the exponent, and it tells you how many times the base number is multiplied.

Understanding how exponentiation works is fundamental when dealing with algebraic expressions because it allows us to manipulate and simplify expressions where variables are raised to a power. With this in mind, remember that \( x^1 = x \), and any number to the power of 0 is 1, meaning \( x^0 = 1 \). These are essential rules when it comes to simplifying expressions with exponents.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations that include addition, subtraction, multiplication, division, and exponentiation. For instance, the expression \( 8x^3y^2 \) includes a constant (8), variables (x and y), and exponents (3 and 2).

When dealing with algebraic expressions, it's important to understand the terms involved. A term is a single entity within an expression that may consist of a constant, a variable, or a combination of constants and variables multiplied together. For example, in the expression \( 8x^3y^2 \), there is only one term. Expressions can be simplified, expanded, factored, or manipulated to solve algebraic equations, done according to the rules of algebra. These rules help ensure that we can work with expressions in a consistent and logical way.

Simplifying Exponents

Simplifying exponents refers to the process of reducing expressions with exponents to their simplest form. This often involves applying the laws of exponents, which include the product rule (multiplying same bases), the quotient rule (dividing same bases), the power rule (raising a power to another power), and the zero exponent rule among others.

For example, in the final step of our initial problem, we simplified the expression \( 8 \times (x \times x \times x) \times (y \times y) \) by removing the parentheses, which is allowed by the associative property of multiplication. We were left with an expanded expression: \( 8x^3y^2 = 8xxxyy \), where the exponents no longer appear explicitly, but the expression demonstrates the repeated multiplication indicated by the original exponents.

Remember, the goal of simplifying is not always to remove the exponents entirely but to create an expression that is easier to work with, whether that means expressing it in a more concise form or expanding it to reveal a particular pattern or simplify further operations.