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Chapter 4: Problem 60

Simplify each expression. Assume no base is \(0 .\) $$y^{m+1} x^{m}$$

Short Answer

Expert verified
The simplified form of the expression is \(y^{m+1} x^{m}\).

Step by step solution

01

Understand the problem

The given task is to simplify the expression \(y^{m+1} x^{m}\). It looks like a multiplication operation of \(y^{m+1}\) and \(x^{m}\).
02

Apply the laws of exponents

As this problem involves powers, additionally, the bases are different (y and x) and hence cannot be combined. Nothing further can be done, so that \(y^{m+1} x^{m}\) is already simplified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laws of Exponents
The laws of exponents are fundamental rules that simplify expressions involving powers. They help in efficiently dealing with expressions that have exponents by providing shortcuts. Here are some important laws:
  • Product of Powers: When multiplying two exponents with the same base, you add the exponents \(a^m \times a^n = a^{m+n}\).
  • Power of a Power: If an exponent is raised to another power, you multiply the exponents \((a^m)^n = a^{m \cdot n}\).
  • Zero Exponent: Any non-zero base raised to the power of zero equals 1 \(a^0 = 1\), given \(a eq 0\).
  • Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the opposite positive power \(a^{-n} = \frac{1}{a^n}\).
In the original exercise, we have different bases \(y\) and \(x\). Since the exponents require the same base to apply these rules, no further simplification can occur. But knowing these laws is essential for solving more complex expressions.
Simplification
Simplification is all about rewriting mathematical expressions in a simpler, more compact form without changing their value. It often involves combining like terms, reducing fractions, or applying mathematical laws.
  • Like Terms: Combine terms that have the same variables and exponent power.
  • Reduce: Use known identities to make expressions more concise (e.g., using the laws of exponents).
  • Evaluate: Substitute values if possible to give a numerical answer.
In our exercise with expression \(y^{m+1} x^{m}\), we see that the bases \(y\) and \(x\) do not share commonality in terms. Thus, no further simplification beyond the original expression is possible. Remember, the aim is clarity - restating expressions in a clearer, more direct form where possible.
Variables and Exponents
Variables and exponents are key components of algebraic expressions. A variable is a symbol used to represent a number, typically \(x\), \(y\), or \(z\). Exponents indicate how many times a base is multiplied by itself.In algebra:
  • Variables: Serve as placeholders that can change or are unknown.
  • Exponents: Determine the "degree" or "power" by which a variable is raised.
In the problem \(y^{m+1} x^{m}\), \(y\) and \(x\) are variables, while \(m+1\) and \(m\) are their exponents. The higher the exponent, the more times a variable is repeated in multiplication. Handling these correctly is crucial as they form the backbone of algebraic attempts at simplification and expansion in equations.

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