Question

Without a calculator, decide whether the quantities are positive or negative. $$ (-61)^{-42} $$

Short answer

Based on the analysis and solution provided, we can conclude that the given expression \((-61)^{-42}\) is positive.

Step-by-step solution

Understand the expression

The given expression \((-61)^{-42}\) means that we have to evaluate \((-61)\) raised to the power of \(-42\). In this case, the base is \(-61\), and the exponent is \(-42\).

Determine the sign of the result using the exponent's properties

Using the properties of exponents, we know that if the exponent is negative, we can write the expression as the reciprocal of the base raised to the positive exponent. So, we have: $$ (-61)^{-42} = \frac{1}{(-61)^{42}} $$ Now, since the exponent is positive and even, the result of an even power of a negative number is positive. Therefore, we have: $$ (-61)^{42} > 0 $$

Final statement

Since \((-61)^{42}\) is positive, its reciprocal, \((-61)^{-42}=\frac{1}{(-61)^{42}}\), is also positive. Hence, the quantity in the given expression is positive.

Key concepts

Negative Exponents

Many students find negative exponents challenging at first, but understanding them can make math a lot easier. A negative exponent means you take the reciprocal of the base. Here, the exponent (-42) is negative, so you rewrite the expression as the reciprocal. This means:

• An expression of the form \(a^{-n}\) is equivalent to \(\frac{1}{a^{n}}\).
• This flips the base to the denominator when the exponent is negative.

The goal is to convert the negative exponent into a positive one, which makes calculations simpler and straightforward. The basic rule to remember is if the exponent is negative, flip the base to the opposite side of the fraction line and change the exponent to positive. In our exercise, \((-61)^{-42}\) becomes \(\frac{1}{(-61)^{42}}\). This transformation helps in evaluating the expression efficiently.

Properties of Exponents

Understanding the properties of exponents is crucial in evaluating expressions correctly. These properties provide useful shortcuts for simplifying expressions:

• Multiplying with the same base: \(a^m \times a^n = a^{m+n}\).
• Dividing with the same base: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a power: \((a^m)^n = a^{m \times n}\).
• Zero exponent rule: \(a^0 = 1\), for any non-zero \(a\).
• Negative exponents: \(a^{-n} = \frac{1}{a^n}\).

In our example, the property of negative exponents simplifies \((-61)^{-42}\) to \(\frac{1}{(-61)^{42}}\). This tells us that the operation performed by the exponent is not just about repeating multiplication, but also involves understanding whether the result changes its position in a fraction when negative exponents are involved.

Reciprocal of a Number

The term 'reciprocal' might sound sophisticated, but it's simple in practice. The reciprocal of a number is what you multiply it by to get a product of 1. For any non-zero number \(a\), the reciprocal is \(\frac{1}{a}\). Here are some quick hints about reciprocals:

• They're essential in dealing with division and negative exponents.
• If \(x\) is a number, its reciprocal is \(\frac{1}{x}\).
• The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).

In the given expression \((-61)^{-42}\), turning it into its reciprocal using \(\frac{1}{(-61)^{42}}\) helps in simplifying and evaluating. Using the reciprocal makes it easier to handle complex problems involving negative exponents, effectively turning division problems into multiplication problems by inverting the terms.