Question

Remove parentheses and simplify. $$(0.826 c-5.37)-(2.76+0.273 c)$$

Short answer

The simplified expression is \(0.553c - 8.13\).

Step-by-step solution

Distribute the Negative Sign

To remove the parentheses, distribute the negative sign to the terms inside the second set of parentheses. This involves changing the signs of those terms.

Combine Like Terms

After distributing the negative sign, combine like terms by adding or subtracting the coefficients of the terms that contain the variable 'c' and then combine the constant terms.

Simplify the Expression

Once the like terms are combined, simply write down the simplified expression, which is the result.

Key concepts

Combining Like Terms

When working with algebraic expressions, one of the fundamental techniques is 'combining like terms'. Like terms are terms that have the same variable raised to the same power. The only thing that can differ is their coefficients, the numbers in front of the variables. For instance, in the expression \(2x + 3x\), the terms \(2x\) and \(3x\) are like terms since they both contain the variable \(x\) raised to the first power.

To combine these like terms, simply add or subtract the coefficients and keep the variable part unchanged. So, \(2x + 3x\) would simplify to \(5x\). This process makes the expression more concise and easier to work with. It's important to carry out this step carefully to avoid errors in the final result.

Distributive Property

The 'distributive property' is a critical concept that allows us to multiply a single term by each term within a parenthesis in an algebraic expression. For instance, if we have \(a(b + c)\), the distributive property allows us to rewrite this as \(ab + ac\), distributing the multiplication of \(a\) to each term inside the parentheses.

This property also applies to the distribution of a negative sign, such as in the problem \((0.826 c-5.37)-(2.76+0.273 c)\). To simplify this expression, we must distribute the negative sign across the terms in the second parenthesis, effectively changing the addition to subtraction, resulting in \(-2.76 -0.273c\). Understanding this principle is key to manipulating and simplifying expressions correctly.

Algebraic Simplification

The process of 'algebraic simplification' combines several steps to transform a complex expression into its simplest form. It often includes distributing negative signs, combining like terms, and performing any necessary arithmetic.

Let's consider the expression given in the exercise. After distributing the negative sign, we would have \(0.826c - 5.37 - 2.76 - 0.273c\). The next step is to combine like terms which, in this case, are \(0.826c\) and \(-0.273c\) for the variables, and \(-5.37\) and \(-2.76\) for the constants. By performing the addition or subtraction as necessary, we simplify the expression to find the most reduced form.

Algebraic simplification aims to present expressions in the clearest and most concise manner, which not only makes them easier to understand but also easier to use in further calculations or applications.