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Chapter 6: Problem 15

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$\sqrt{7} \sqrt{7}$$

Short Answer

Expert verified
The power simplifies to 7.

Step by step solution

01

Identify the Expression

The given expression is \( \sqrt{7} \times \sqrt{7} \). This can be simplified using the property of square roots that states \( \sqrt{a} \times \sqrt{a} = a \).
02

Simplify the Expression

Using the property from Step 1, the expression \( \sqrt{7} \times \sqrt{7} \) simplifies to 7. Thus, mathematically, \( \sqrt{7} \times \sqrt{7} = 7 \).
03

Verify Using a Calculator

Although we already simplified, you may still want to verify using a calculator: Enter \( \sqrt{7} \)*\(\sqrt{7}\). The calculator should confirm that it equals 7 with no additional decimal places.

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Key Concepts

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

Calculator Usage in Mathematics
Using calculators in mathematics can greatly enhance both the understanding and efficiency of solving problems. Whether you're calculating simple arithmetic problems or performing more complex operations like square roots, calculators can provide quick and accurate results.
Here鈥檚 how you can use a calculator to find the square root of a number like 7:
  • First, turn on your calculator.
  • Enter the number 7.
  • Press the square root button (usually represented by \(\sqrt{}\) ).
  • To find \( \sqrt{7} \times \sqrt{7} \), simply re-enter the number or use the multiplication function.
  • The calculator will then show the result as 7, confirming the mathematical simplification.
Using a calculator is a straightforward process that helps verify results and ensures that mathematical calculations are correct. Remember, when using a calculator, to check the mode is set correctly for scientific calculations if needed.
Mathematical Properties
Mathematical properties are foundational truths that apply throughout various branches of mathematics. They simplify expressions and solve equations effectively. One such property is related to square roots:
  • Property of Product of Square Roots: \( \sqrt{a} \times \sqrt{a} = a \). This property lets us simplify expressions involving squares quickly.
In our example of \( \sqrt{7} \times \sqrt{7} \), this property directly tells us the result is 7.
Understanding these properties helps avoid misunderstandings and errors in solving problems. It allows you to see patterns and connect different areas of mathematics intuitively. You can confidently reason through complex mathematical equations using a solid grasp of fundamental properties.
Simplification of Expressions
Simplifying expressions reduces them to their simplest form. This process can make solving problems easier by focusing on core components rather than unnecessary complexity. For square roots, this involves recognizing and applying relevant properties:
  • Identify repeated elements or factors within an expression. For instance, \( \sqrt{7} \times \sqrt{7} \).
  • Use recognized mathematical properties, such as the property of the square root product \( \sqrt{a} \times \sqrt{a} = a \).
  • Write down or calculate the simplified result, simplifying the computation to its most basic form.
In our case, the simplification of \( \sqrt{7} \times \sqrt{7} \) to 7 avoids the need for further calculations and shows directly how the expression resolves. Simplification helps confirm the validity of complex operations by reducing an expression to one of its simplest, truest forms.

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Most popular questions from this chapter

Salinity The salinity of the oceans changes with latitude and depth. In the tropics, the salinity increases on the surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by $$f(x)=31.5+1.1 \log (x+1)$$ models salinity to depths of 1000 meters at a latitude of 57.5". The variable x is the depth in meters, and \(f(x)\) is in grams of salt per kilogram of seawater. (Source: Hartman, \(D .\) Global Physical Climatology, Academic Press.) Approximate analytically, to the nearest meter, the depth where the salinity equals 33.

Use a graph with the given viewing window to decide which functions are one- to-one. If a function is one-to-one, give the equation of its inverse function. Check your work by graphing the inverse function on the same coordinate axes. $$f(x)=\frac{-x}{x-4} ;[-2.6,10.6] \text { by }[-4.1,4.1]$$

Use any method (analytic or graphical) to solve each equation. $$\ln \left(\ln e^{-x}\right)=\ln 3$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$2 \log _{m} a-3 \log _{m} b^{2}$$

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=3 x-4$$
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