91Ó°ÊÓ

In algebra, substitution is a fundamental skill where you replace a variable with its numerical value. It's like swapping out a placeholder with its actual content. Consider you are hosting a dinner party and have place cards for each guest. When a guest arrives, you replace their placeholder card with them in person. Similar to our dinner party, in algebra, if you know that a variable, such as \(a\), represents a number, like -1, you substitute -1 in for every instance of \(a\).

For example, with the expression \(-3a^{33}\), we follow these steps:

• Determine the value of the variable, which is given: \(a=-1\).
• Replace the variable \(a\) in the expression with its value: \(-3(-1)^{33}\).

This essential method allows us to move towards a numerical solution rather than a symbolic one.

When dealing with exponents, the goal is often to simplify the expression to make it easier to understand and solve. Simplifying an exponent means reducing it to its most basic form. A base raised to an exponent tells you how many times to multiply the base by itself. For instance, \(2^3\) is \(2 \times 2 \times 2\) which equals 8.

In our exercise, we encounter an exponent in the form of a negative base raised to an odd power: \((-1)^{33}\). Here's what simplifying involves in this case:

• Recognize that an odd power of -1 will always give -1.
• Knowing this, we can immediately say that \((-1)^{33} = -1\), without the need for further calculation.

Understanding the properties of exponents, like the fact negative numbers to odd powers remain negative, helps simplify expressions efficiently.

Multiplying integers may seem straightforward, but it involves understanding the signs of the numbers as well as their magnitudes. When multiplied together, the sign of the result depends on the signs of the integers involved. If you multiply two integers with the same sign, whether both are positive or both are negative, the result is always positive. However, if the integers have different signs, the result is negative.

In our example, after substitution and simplifying the exponents, we’re left with the multiplication problem \(-3 \times -1\). Since both integers are negative, their product is positive. So, \(-3 \times -1 = 3\). This aligns with the general rule:

• \(negative \times negative = positive\)
• \(positive \times positive = positive\)
• \(negative \times positive = negative\)
• \(positive \times negative = negative\)

Mastering the rules for multiplying integers is a critical step in solving algebraic expressions correctly.