Problem 39 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ -(3 x)^{2} \cdot\left(7 x^{4}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-441x^{10}\).

Step 1: Apply the power to each element inside the brackets

For \(-(3x)^{2}\) , square both 3 and \(x\): \((3^{2})(x^{2}) = 9x^{2}\). \nFor \((7x^{4})^{2}\), both 7 and \(x^{4}\) are raised to the power of 2: \( (7^{2})(x^{4 \cdot 2}) = 49x^{8}\).

Step 2: Multiply the two results obtained in Step 1

Multiply \(9x^{2}\) by \(49x^{8}\) to get \(441x^{10}\). Note that when multiplying terms with the same base (in this case, \(x\)), you add the exponents. Hence, \(x^{2} \cdot x^{8} = x^{10}\).

Step 3: Apply the initial negative sign

Finally, apply the initial negative sign to the result, giving \(-441x^{10}\).

Key Concepts

Exponent Rules

When dealing with exponents in algebra, there are several key rules that simplify the process of working with powers. One fundamental rule is the power of a power rule, which states that \( (a^n)^m = a^{n \cdot m} \). This means that when you raise a power to another power, you multiply the exponents. Similarly, the product of powers rule applies when multiplying two exponents with the same base: \( a^n \cdot a^m = a^{n+m} \). In this case, you add the exponents.

Applying these exponent rules helps in simplifying expressions effectively. For example, attacking a complex problem like \( -(3x)^2 \cdot (7x^4)^2 \) becomes manageable when we break it down: \( (3x)^2 \) results in \( 3^2x^2 \) and \( (7x^4)^2 \) simplifies to \( 7^2x^{4 \cdot 2} \), utilizing the aforementioned exponent rules. Subsequently, the product of these simplified terms involves yet another exponent rule, the product of powers rule, leading us to the final expression.

Multiplying Polynomials

Multiplying polynomials involves combining like terms and applying exponent rules. When polynomials consist of the same base variables, we can use exponent rules to combine them efficiently. Consider the form \( (ax^n) \cdot (bx^m) \), where 'a' and 'b' are coefficients, and 'n' and 'm' are exponents: the combined term will be \( ab \cdot x^{n+m} \).

Let's incorporate this understanding into the exercise at hand: when multiplying \( 9x^2 \) by \( 49x^8 \), we are essentially multiplying the coefficients (9 and 49) and adding the exponents of 'x' (2 and 8) resulting in \( 441x^{2+8} \) or \( 441x^{10} \). This clear-cut approach not only simplifies the multiplication of polynomials but also ensures accuracy in the process.

Negative Numbers in Algebra

Working with negative numbers in algebra requires careful attention to signs. A negative sign can represent the opposite of a number or be indicative of subtraction. In expressions, a negative sign in front of a parenthesized term applies to the entire term once the parentheses are removed or the operation within is carried out.

For instance, in the expression \( -(3x)^2 \cdot (7x^4)^2 \), the initial negative sign in front of \( (3x)^2 \) means that after simplifying \( (3x)^2 \) to \( 9x^2 \) according to exponent rules, this negative sign will then apply to the product of \( 9x^2 \) and \( 49x^8 \) to yield \( -441x^{10} \). Remember, a negative times a positive is always a negative, and this rule ensures that the sign is correctly applied to the final result in algebraic operations.