Question

MENTAL MATH Write a question that could be used to solve the equation. Then use mental math to solve the equation. $$y^{3}=125$$

Short answer

The solution to the equation \(y^3 = 125\) is \(y = 5\).

Step-by-step solution

Recognize the Equation

The given equation is \(y^3 = 125\). The problem can be resolved by simply finding the cube root of 125. The cube root of a number is a special value that, when used in a multiplication three times, gives that number.

Solve for y

In mental math, the cube root of 125 can be identified as 5 because when 5 is multiplied by itself three times (5*5*5), the result is 125. So, \(y^3 = 125\) can be rewritten as \(y = \sqrt[3]{125}\), which simplifies to \(y = 5\).

Key concepts

Mental Math

When it comes to solving equations, mental math refers to the ability to perform calculations in your head without the need for a calculator or paper. It's a skill that is handy for quick estimations or when working with simple numbers. For the exercise at hand, finding the cube root of 125 using mental math requires recognizing that this is finding a number that, when multiplied by itself three times, equals 125.

For many, the process of finding a cube root mentally involves recalling perfect cubes or using estimation. Perfect cubes are numbers like 1, 8, 27, 64, and so on, which are produced by a whole number multiplied by itself twice more. Knowing perfect cubes up to a specific range can help you quickly identify the cube root of a number that is a perfect cube, like 125.

One way to improve at mental math is to practice the factorization of small numbers and to memorize perfect squares and cubes up to at least 12^3. The more you familiarize yourself with these numbers, the easier it will be to solve cubic equations mentally.

Solving Cubic Equations

Cubic equations are polynomial equations of the third degree, which means the highest exponent of the variable is three. The general form of a cubic equation is \(ax^3 + bx^2 + cx + d = 0\). However, not all cubic equations come in this expanded form; some can be simplified, like in the given exercise \(y^3 = 125\), which is already set equal to a constant.

To solve a cubic equation, you can factor the equation or use methods such as synthetic division or the Rational Root Theorem when applicable. In simpler cases, like the given exercise, recognizing perfect cubes can allow for immediate factorization and solution. When a cubic equation cannot be factored easily, more complex methods like Cardano's method may be required.

For educational purposes, starting with equations that involve perfect cubes helps students understand the structure of cubic equations and builds a foundation for tackling more complex problems. As proficiency grows, students can be introduced to cubic equations where coefficients need to be considered.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent signifies how many times the base is multiplied by itself. For instance, \(5^3\) means 5 multiplied by itself two additional times (since the exponent is 3), which equals 125.

Understanding exponentiation is fundamental when dealing with equations involving powers, like quadratic and cubic equations. For cubic equations, like the one you are solving, \(y^3 = 125\), recognizing that you are dealing with a base raised to the power of 3 can immediately tell you that you are looking for a cube root.

In teaching exponentiation, it is important to emphasize the relationship between exponentiation and its inverse operations, roots. Highlighting the connection between squaring a number and taking a square root, or cubing a number and finding its cube root, aids in demystifying more advanced topics and ensures that students have a solid grasp of how these operations interrelate.