Exponents comprise a juicy tidbit of basic-math-facts material. Exponents allow us to raise numbers, variables, and even expressions to powers, thus achieving repeated multiplication. The ever present exponent in all kinds of mathematical problems requires that the student be thoroughly conversant with its features and properties. Here we look at the laws, the knowledge of which, will allow any student to master this topic. ™ In the expression 3^2, which is read "3 squared," or "3 to the second power," 3 is the base and 2 is the power or exponent. The exponent tells us how many times to use the base as a factor. The same applies to variables and variable expressions. In x^3, this mean x*x*x. In (x + 1)^2, this means (x + 1)*(x + 1). Exponents are omnipresent in algebra and indeed all of mathematics, and understanding their properties and how to work with them is extremely important. Mastering exponents requires that the student be familiar with some basic laws and properties. Product Law When multiplying expressions involving the same base to different or equal powers, simply write the base to the sum of the powers. For example, (x^3)(x^2) is the same as x^(3 + 2) = x^5. To see why this is so, think of the exponential expression as pearls on a string. In x^3 = x*x*x, you have three x's (pearls) on the string. In x^2, you have two pearls. Thus in the product you have five pearls, or x^5. Quotient Law When dividing expressions involving the same base, you simply subtract the powers. Thus in (x^4)/(x^2) = x^(4-2) = x^2. Why this is so depends on the cancellation property of the real numbers. This property says that when the same number or variable appears in both the numerator and denominator of a fraction, then this term can be canceled. Let us look at a numerical example to make this completely clear. Take (5*4)/4. Since 4 appears in both the top and bottom of this expression, we can kill it---well not kill, we don't want to get violent, but you know what I mean---to get 5. Now let's multiply and divide to see if this agrees with our answer: (5*4)/4 = 20/4 = 5. Check. Thus this cancellation property holds. In an expression such as (y^5)/(y^3), this is (y*y*y*y*y)/(y*y*y), if we expand. Since we have 3 y's in the denominator, we can use those to cancel 3 y's in the numerator to get y^2. This agrees with y^(5-3) = y^2. Power of a Power Law In an expression such as (x^4)^3, we have what is known as a power to a power. The power of a power law states that we simplify by multiplying the powers together. Thus (x^4)^3 = x^(4*3) = x^12. If you think about why this is so, notice that the base in this expression is x^4. The exponent 3 tells us to use this base 3 times. Thus we would obtain (x^4)*(x^4)*(x^4). Now we see this as a product of the same base to the same power and can thus use our first property to get x^(4 + 4+ 4) = x^12. Distributive Property This property tells us how to simplify an expression such as (x^3*y^2)^3. To simplify this, we distribute the power 3 outside parentheses inside, multiplying each power to get x^(3*3)*y^(2*3) = x^9*y^6. To understand why this is so, notice that the base in the original expression is x^3*y^2. The 3 outside parentheses tells us to multiply this base by itself 3 times. When you do that and then rearrange the expression using both the associative and commutative properties of multiplication, you can then apply the first property to get the answer. Zero Exponent Property Any number or variable---except 0---to the 0 power is always 1. Thus 2^0 = 1; x^0 = 1; (x + 1)^0 = 1. To see why this is so, let us consider the expression (x^3)/(x^3). This is clearly equal to 1, since any number (except 0) or expression over itself yields this result. Using our quotient property, we see this is equal to x^(3 - 3) = x^0. Since both expressions must yield the same result, we get that x^0 = 1. Negative Exponent Property When we raise a number or variable to a negative integer, we end up with the reciprocal. That is 3^(-2) = 1/(3^2). To see why this is so, let us consider the expression (3^2)/(3^4). If we expand this, we obtain (3*3)/(3*3*3*3). Using the cancellation property, we end up 數學補習 dse with 1/(3*3) = 1/(3^2). Using the quotient property we that (3^2)/(3^4) = 3^(2 - 4) = 3^(-2). Since both of these expressions must be equal, we have that 3^(-2) = 1/(3^2). Understanding these six properties of exponents will give students the solid foundation they need to tackle all kinds of pre-algebra, algebra, and even calculus problems. Often times, a student's stumbling blocks can be removed with the bulldozer of foundational concepts. Study these properties and learn them. You will then be on the road to mathematical mastery. Most children consider math to be the hardest subject in school. Their case may be merely an emotional one. However, we have to investigate the reasons behind this widespread hatred of math in children. If your child does not like math, these can be the following reasons for your child's dislike; (1) inability to comprehend the basics of math, (2) natural aptitude against math, (3) boring material and syllabus for math classes, (4) uninspiring teaching methods, (5) no interest of the parents. Any one or all of these reasons can make math the hardest subject for a child. If you do not develop a liking in your child for math during pre-school , your child may not be able to develop a natural aptitude for math. Kids like to play with toys, play various interesting little games and making paintings or the like. Parents should try to teach the child about counting things and the basic concepts in their routine activities. This does not make it a boring task and your child develops a natural aptitude for math. This will solve another problem for your child as well. This will develop a very strong base for math in your child and when he will go to school he will not be behind the other students. Having developed an aptitude for math, your child will perform better in his subject. Another problem that makes math the hardest subject in school is an inefficient and boring math syllabus. The syllabus should be developed while keeping the level and the liking of children. They find math to be boring and disgusting because syllabus books present it in a dry and boring manner. The next cause that makes math the hardest subject in school is the teaching method. If the teacher himself or herself does not like math, how could she be able to generate an interest in the subject for your child? Teachers do not give examples from every day life. The children start to assume that math does not have any importance in their lives. Then why should they study a subject that has no utility in real life? This is why children start losing interest in math. To make math an interesting subject, teachers have to make it a living subject with examples from the every day life of children. Most of the teachers focus all their energies to prevent mistakes and the creative abilities of children go unnoticed for the most part. This method creates a negative impression about math. If we take away inventiveness from the subject then boring addition or multiplication makes math the hardest subject in school. Children may develop a distaste for math in the long run. There is another major factor that makes math the hardest subject for your child. Mostly parents do not give proper attention and encouragement to children. You may not be able to solve syllabus or teaching method problems. However you can save your child from failure in math at your own home. Schools can not give attention to each child; the teachers have to teach to the whole class. Parents can give attention to their child's problems. You can make math an interesting subject for your child by playing math games or assigning amusing math tasks. The homework routine should be made fun for the child. You must take interest in your child's homework and help her wherever she needs it. If you find it difficult to give the necessary help to your child, you can hire someone who can make math an interesting subject for your child.
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