Question

. Find the value of each of the following; if undefined, say so. (a) \(0 \cdot 0\) (b) \(\frac{0}{0}\) (c) \(\frac{0}{17}\) (d) \(\frac{3}{0}\) (e) \(0^{5}\) (f) \(17^{0}\)

Short answer

(a) 0; (b) undefined; (c) 0; (d) undefined; (e) 0; (f) 1

Step-by-step solution

Multiplication of Zero

When multiplying zero by another zero, the result is always zero. So, \(0 \cdot 0 = 0\).

Undefined Division

Division by zero is undefined in mathematics because any number divided by zero does not produce a meaningful result. Thus, \( \frac{0}{0} \) is undefined.

Zero Divided by a Non-Zero Number

Dividing zero by any non-zero number results in zero because zero divided by any number remains zero. Therefore, \( \frac{0}{17} = 0 \).

Division by Zero

Dividing any number by zero is undefined. Therefore, \( \frac{3}{0} \) is undefined.

Zero Raised to a Positive Power

Any non-zero number raised to a positive power remains zero. Therefore, \( 0^{5} = 0 \).

Any Number Raised to the Power of Zero

Any non-zero number raised to the power of zero is one. Therefore, \( 17^0 = 1 \).

Key concepts

Multiplication of Zero

When you encounter a problem that involves multiplication with zero, it's important to remember that zero has a unique property. No matter what you multiply by zero, the result is always zero. This happens because zero represents nothing. Even if you have zero groups of an object, or you multiply an object zero times, you still end up with nothing. For example, the expression \(0 \cdot 0\) equals zero because you have zero multiplied by zero, which leaves you with zero.

Whenever you see zero in a multiplication equation, you can quickly conclude that the product will be zero. Here are key points to remember about multiplication involving zero:

• Zero times any number equals zero: \( 0 \cdot n = 0 \).
• Any number times zero equals zero: \( n \cdot 0 = 0 \).

These properties make working with zero in multiplication straightforward and simple.

Undefined Division

When dividing numbers, we hope to find a meaningful quotient. However, division by zero presents a unique challenge in mathematics. It's important to know why division by zero is considered undefined. When you attempt to divide any number by zero, such as in the expression \( \frac{3}{0} \), you are asking how many groups of zero fit into the number. This might seem paradoxical because dividing by zero suggests creating groups of nothing, which isn't practically or theoretically possible.

There are two situations to understand here:

• If the dividend (the number being divided) is zero, like in \( \frac{0}{0} \), this seems equally undefined because zero divided by zero suggests anywhere from none to infinitely many solutions.
• Any non-zero number divided by zero, such as \( \frac{3}{0} \), is simply undefined as no quotient makes sense in practical or theoretical terms.

Thus, it’s vital to remember that any time you see a number divided by zero, the result is undefined.

Zero Exponent Rule

Exponents can seem tricky, but the zero exponent rule is straightforward once you understand it. This rule states that any non-zero number raised to the power of zero equals one. This might seem unintuitive at first; however, it makes sense when you consider the pattern of decreasing exponents. Take, for example, the power of two:

• \( 2^3 = 8 \)
• \( 2^2 = 4 \)
• \( 2^1 = 2 \)
• \( 2^0 = 1 \)

As you divide by the base (2 in this case) each time you decrease the exponent, the pattern results in one when the exponent reaches zero. This explains why expressions like \( 17^0 \) evaluate to one.

However, it is key to emphasize that this rule doesn't apply if the base is zero. So expressions like \( 0^5 \) simply result in zero because, as previously discussed, zero remains zero when raised to any positive power. Thus, understanding the zero exponent rule is essential for mastering basic arithmetic calculations.