Question

Show that \(x^{3}>y^{3}\) if \(x>y\)

Short answer

To show that \(x^3 > y^3\) if \(x > y\), we recall the property of inequalities: if \(a > b\) and \(b > 0\), raising both sides to a positive power preserves the inequality. Since \(x > y\), raising both sides to the power of 3 gives us \(x^3 > y^3\), as required.

Step-by-step solution

Recall properties of inequalities

We know that if a > b, and b > 0, then raising both sides to a positive power preserves the inequality. In our case, x > y, and we will be raising both sides to the power of 3 (a positive number), so the inequality will still hold.

Raise both sides to the power of 3

Given x > y, now raise both sides to the power of 3. By the property of inequalities mentioned in step 1, we will still have x³ > y³. So, when we cube both sides, we get: \(x^3 > y^3\) This shows that x³ > y³ if x > y, as required.

Key concepts

Exponents

Exponents are a powerful mathematical tool that allows us to express repeated multiplication in a simplified form. When you see a number raised to an exponent, it means that you multiply that number by itself a certain number of times. For instance, in the expression \(x^3\), \(x\) is multiplied by itself three times: \(x \times x \times x\). Exponents are useful because they help to simplify expressions and can make calculations easier to handle.

There are some basic rules to remember when working with exponents, such as:

• Any number raised to the power of one is itself: \(x^1 = x\).
• Any number raised to the power of zero is one: \(x^0 = 1\), provided \(x eq 0\).
• When you multiply numbers with the same base, you add the exponents: \(x^a \times x^b = x^{a+b}\).

Understanding these rules can make working with exponents easier and prevent common mistakes.

Properties of Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating that one value is greater or less than the other. The key to dealing with inequalities is understanding that they have rules similar to those of equations, but with a few differences.

When you perform operations on both sides of an inequality, you must be careful to maintain the inequality's direction. The basic properties include:

• Adding or subtracting the same value from both sides of an inequality does not change its direction. For example, if \(a > b\), then \(a + c > b + c\).
• Multiplying or dividing both sides by a positive number maintains the inequality's direction. For example, if \(a > b\) and \(c > 0\), then \(ac > bc\).
• Multiplying or dividing both sides by a negative number reverses the inequality's direction. For instance, if \(a > b\) and \(c < 0\), then \(ac < bc\).

These properties are crucial for solving and transforming inequalities correctly.

Cubing Numbers

Cubing a number means raising it to the power of three. When you cube a number, you multiply it by itself twice more, which is different from merely squaring it. For instance, cubing the number \(x\) involves calculating \(x \times x \times x\), resulting in \(x^3\).

Cubing is a particular application of exponents and is often used in mathematical problems and real-world applications alike. It's important to understand that cubing a negative number results in a negative number because the product of three negative factors is negative. For example, \((-2)^3 = -2 \times -2 \times -2 = -8\).

Cubing can lead to larger numbers very quickly, making it a powerful operation. It is also important to remember that cubing positive numbers always yield positive results, which can help simplify inequality problems involving cubed terms.